Introduction to Economic and Strategic Behavior Introduction LeventKo¸ckesen Ko¸cUniversity LeventKo¸ckesen (Ko¸cUniversity) Introduction 1 / 19 Description and Objectives Description Analysis of strategic interactions among individuals using game theory Applications to economics, business, politics, law, biology, history Evolution of social norms and other social institutions Learning Objectives Recognize real strategic situations to which game theory can be applied Model strategic situations as games Apply tools of game theory to gain insights about real life situations View social and economic issues from a completely new perspective LeventKo¸ckesen (Ko¸cUniversity) Introduction 2 / 19 Game Theory: Definition and Assumptions Game theory studies strategic interactions within a group of individuals ◮ Actions of each individual have an effect on the outcome ◮ Individuals are aware of that fact Individuals are rational ◮ have well-defined objectives over the set of possible outcomes ◮ implement the best available strategy to pursue them Rules of the game and rationality are common knowledge LeventKo¸ckesen (Ko¸cUniversity) Introduction 3 / 19 Example 10 people go to a restaurant for dinner Order expensive or inexpensive fish? ◮ Expensive fish: value = 18, price = 20 ◮ Inexpensive fish: value = 12, price = 10 Everbody pays own bill ◮ What do you do? ◮ Single person decision problem Total bill is shared equally ◮ What do you do? ◮ It is a GAME LeventKo¸ckesen (Ko¸cUniversity) Introduction 4 / 19
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Introduction to Economic and Strategic BehaviorIntroduction
Game theory forces you to see a business situation over many periodsfrom two perspectives: yours and your competitors
◮ Judy Lewent - CFO, Merck
To help predict competitor behavior and determine optimal strategy,our consulting teams use techniques such as pay-off matrices andcompetitive games
◮ John Stuckey & David White - McKinsey
At Bell Atlantic, we’ve found that the lessons of game theory give usa wider view of our business situation and provide us a more nimbleapproach to corporate planning
You should try to predict what the other players are going to do◮ Who are the other players?◮ What are their possible moves?◮ What do they want from the game?◮ What do they know about the game?
Modeling the strategic interaction correctly is very important
Introduction to Economic and Strategic BehaviorStrategic Form Games
Levent Kockesen
Koc University
Levent Kockesen (Koc University) Strategic Form Games 1 / 1
Contribution Game
Everybody starts with 10 TL
You decide how much of 10 TL to contribute to joint fund
Amount you contribute will be doubled and then divided equallyamong everyone
I will distribute slips of paper that looks like this
Name:
Your Contribution:
Write your name and an integer between 0 and 10
We will collect them and enter into Excel
We will choose one player randomly and pay herClick here for the EXCEL file
Levent Kockesen (Koc University) Strategic Form Games 2 / 1
Contribution Game
Who are the other players?
What are their possible moves?
What do they want from the game?
What do they know about the game?
Levent Kockesen (Koc University) Strategic Form Games 3 / 1
Strategic Form Games
It is used to model situations in which players choose strategieswithout knowing the strategy choices of other players
Also known as normal form games
A strategic form game is composed of
1. Set of players
2. A set of strategies for each player
3. A payoff function for each player
An outcome is a collection of strategies, one for each player.◮ Also known as a strategy profile
Levent Kockesen (Koc University) Strategic Form Games 4 / 1
Price Competition
You will be matched randomly with another student
Each will independently choose High or Low price
If you both choose High: Each gets 10
If both Low: Each gets 5
If one High the other Low: High gets 2, Low gets 15
Make you choice and write on a piece of paper
Levent Kockesen (Koc University) Strategic Form Games 5 / 1
Price Competition
Player 1
Player 2High Low
High 10, 10 2, 15Low 15, 2 5, 5
Bimatrix
This way of writing the game is known as a bimatrix
Levent Kockesen (Koc University) Strategic Form Games 6 / 1
Price Competition
Player 1
Player 2High Low
High 10, 10 2, 15Low 15, 2 5, 5
What should Player 1 play?
Does that depend on what she thinks Player 2 will do?
Low is an example of a dominant strategy
It is optimal independent of what other players do
How about Player 2?
(Low, Low) is a dominant strategy equilibrium
Levent Kockesen (Koc University) Strategic Form Games 7 / 1
Dominant Strategies
A strategy A strictly dominates another strategy B if it yields a strictlyhigher payoff irrespective of how the other players play.
A weakly dominates B if it never does worse than B and sometimes doesstrictly better.
A strategy A is strictly dominant if it strictly dominates every otherstrategy of that player. It is called weakly dominant if it weakly dominatesevery other strategy.
Levent Kockesen (Koc University) Strategic Form Games 8 / 1
Dominant Strategy Equilibrium
If every player has a (strictly or weakly) dominant strategy, then thecorresponding outcome is a (strictly or weakly) dominant strategyequilibrium.
Levent Kockesen (Koc University) Strategic Form Games 9 / 1
Dominant Strategy Equilibrium
T
WH L
H 10, 10 2, 15L 15, 2 5, 5
L strictly dominates H
(L,L) is a strictly dominantstrategy equilibrium
T
WH L
H 10, 10 5, 15L 15, 5 5, 5
L weakly dominates H
(L,L) is a weakly dominantstrategy equilibrium
Levent Kockesen (Koc University) Strategic Form Games 10 / 1
Dominant Strategy Equilibrium
A reasonable solution concept
It only demands the players to be rational
It does not require them to know that the others are rational too
But it does not exist in many interesting games
Levent Kockesen (Koc University) Strategic Form Games 11 / 1
Prisoners’ Dilemma
Prisoners’ Dilemma
Player 1
Player 2Confess Deny
Confess −5,−5 0,−10Deny −10, 0 −1,−1
How would you play?
Levent Kockesen (Koc University) Strategic Form Games 12 / 1
Case Study: Advertising Ban in Tobacco Industry
1960s: tobacco companies advertise heavily on TV and radio◮ TV ads accounted for more than 80% of total advertising budget
1964: US Surgeon General’s 1964 report finds cigarette smokinghazardous to health
1970: Public Health Cigarette Smoking Act banned all cigarette TVand radio ads
Question
What is the effect on advertising expenditures? On profits?
Levent Kockesen (Koc University) Strategic Form Games 13 / 1
Case Study: Advertising Ban in Tobacco Industry
Model Assumptions
Symmetric oligopolistic industry
Firms choose how much to spend on advertising
Total demand is perfectly inelastic
Advertising Game
PM
RJRAd No
Ad 30, 30 60, 20No 20, 60 50, 50
Levent Kockesen (Koc University) Strategic Form Games 14 / 1
Case Study: Advertising Ban in Tobacco Industry
What Happened:
Advertising expenditures fell sharply in 1971
Profits rose as stock returns for the major tobacco companies reachedabnormal heights
Aggregate sales remained unchanged
After 5 years: Advertising spending began to recover and even exceedpre-ban levels
See Shi Qi, “The Impact of Advertising Regulation on Industry TheCigarette Advertising Ban of 1971,” RAND J. (2013)
Levent Kockesen (Koc University) Strategic Form Games 15 / 1
Rationality
Players have consistent preferences over outcomes◮ Can rank outcomes◮ Payoffs represent rankings
They have beliefs regarding future events◮ Other players’ moves◮ Chance events
On the basis of these beliefs they choose the best strategy◮ The strategy that maximizes their expected payoffs
Levent Kockesen (Koc University) Strategic Form Games 16 / 1
Rationality
Suppose there are 4 possible outcomes: a, b, c, d
Corresponding payoffs: u(a), u(b), u(c), u(d)
and probabilities p(a), p(b), p(c), p(d)
Expected payoff is
p(a)u(a) + p(b)u(b) + p(c)u(c) + p(d)u(d)
Levent Kockesen (Koc University) Strategic Form Games 17 / 1
Risk Aversion
Choose between◮ S: I will give you $100◮ R: I will flip a fair coin. If heads I will give you $200; If tails I will give
you nothing
If you prefer S over R you are risk averse
If you don’t care you are risk neutral
If you prefer R over S you are risk lover
An individual is risk averse is he prefers to receive the expected value of alottery to playing the lottery
Levent Kockesen (Koc University) Strategic Form Games 18 / 1
Risk Aversion
Payoff function (of certain outcome) is
Linear if risk neutral◮ u(x) = x
Concave if risk averse◮ u(x) =
√x
Convex if risk lover◮ u(x) = x2
Levent Kockesen (Koc University) Strategic Form Games 19 / 1
Risk Preferences
b
b
bb
x
u(x)
x1 x2E[x]
Risk averse: u(E[x]) > E[u(x)]
b
b
b
x
u(x)
x1 x2E[x]
Risk neutral: u(E[x]) = E[u(x)]
b
b
b
b
x
u(x)
x1 x2E[x]
Risk lover: u(E[x]) < E[u(x)]
Levent Kockesen (Koc University) Strategic Form Games 20 / 1
Rationality
Rationality does not mean selfishness◮ Payoff functions must include all relevant considerations such as
altruism
It is a consistency requirement
Should be thought as an approximation to reality
In evolutionary games we do not assume rationality◮ Genes govern behavior◮ Those that are successful reproduce faster◮ An evolutionary stable state is the eventual outcome of this dynamics
Levent Kockesen (Koc University) Strategic Form Games 21 / 1
Common Knowledge
We also assume that the game as well as the rationality is commonknowledge
A fact X is common knowledge if everybody knows it
If everybody knows that everybody knows it
If everybody knows that everybody knows that everybody knows it
... ad infinitum
Levent Kockesen (Koc University) Strategic Form Games 22 / 1
A Puzzle
Three girls sitting in a circle
Each is wearing a hat: Red or White
Each can see the others’ hat but not her own
Teacher asks each to guess her hat color◮ This is done sequentially◮ Assume a wrong guess is very costly
In fact all have red
Could any one of them guess correctly?
Suppose now that the teacher says
At least one of you is wearing a red hat
How about now?
Levent Kockesen (Koc University) Strategic Form Games 23 / 1
Price Matching
Wal-Mart: Our Ad Match Guarantee
We’re committed to providing low prices every day. On everything. So ifyou find a lower advertised price on an identical product, tell us and we’llmatch it. Right at the register.
Levent Kockesen (Koc University) Strategic Form Games 24 / 1
Price Matching
Sounds like a good deal for customers
How does this change the game?
Toys“R”us
Wal-MartHigh Low Match
High 10, 10 2, 15 10, 10Low 15, 2 5, 5 5, 5
Match 10, 10 5, 5 10, 10
Is there a dominant strategy for any of the players?
There is no dominant strategy equilibrium for this game
So, what can we say about this game?
Levent Kockesen (Koc University) Strategic Form Games 25 / 1
Price Matching
Toys“R”us
Wal-MartHigh Low Match
High 10, 10 2, 15 10, 10Low 15, 2 5, 5 5, 5
Match 10, 10 5, 5 10, 10
High is weakly dominated and Toys“R”us is rational◮ Toys“R”us should not use High
High is weakly dominated and Wal-Mart is rational◮ Wal-Mart should not use High
Each knows the other is rational◮ Toys“R”us knows that Wal-Mart will not use High◮ Wal-Mart knows that Toys“R”us will not use High◮ This is where we use common knowledge of rationality
Levent Kockesen (Koc University) Strategic Form Games 26 / 1
Price Matching
Therefore we have the following “effective” game
Toys“R”us
Wal-MartLow Match
Low 5, 5 5, 5Match 5, 5 10, 10
Low becomes a weakly dominated strategy for both
Both companies will play Match and the prices will be high
The above procedure is known as Iterated Elimination of DominatedStrategies (IEDS)
To be a good strategist try to see the world from the perspective of yourrivals and understand that they will most likely do the same
Levent Kockesen (Koc University) Strategic Form Games 27 / 1
Dominated Strategies
A “rational” player should never play an strategy when there isanother strategy that gives her a higher payoff irrespective of how theothers play
We call such a strategy a dominated strategy
A strategy A is strictly dominated by another strategy B if B does strictlybetter than A irrespective of how other players play.
A is weakly dominated by B if B never does worse than A and sometimesdoes strictly better.
Levent Kockesen (Koc University) Strategic Form Games 28 / 1
Iterated Elimination of Dominated Strategies
Common knowledge of rationality justifies eliminating dominatedstrategies iteratively
This procedure is known as Iterated Elimination of DominatedStrategies
If every strategy eliminated is a strictly dominated strategy◮ Iterated Elimination of Strictly Dominated Strategies
If at least one strategy eliminated is a weakly dominated strategy◮ Iterated Elimination of Weakly Dominated Strategies
Remember Guessing Game?
Pick a number between 0 and 100
You win if your number is closest to half the average
What do you pick?
Levent Kockesen (Koc University) Strategic Form Games 29 / 1
Iterated Elimination of Dominated Strategies
Half the average can be at most 50
If everybody is rational◮ Nobody should choose higher than 50
If everybody knows that everybody is rational◮ Nobody should choose higher than 25
If everybody knows that everybody knows that everybody is rational◮ Nobody should choose higher than 12.5
Can eliminate all down to zero
Keynes’ Beauty Contest
Number closest to the average wins.
Shiller’s NYT article
Levent Kockesen (Koc University) Strategic Form Games 30 / 1
Envelope Paradox
Ali and Berna has each an envelope with 5, 10, 20, 40, 80, or 160 TL.
One of the envelopes has twice the amount in the other
Each looks into their envelope and independently decide whether theywant to exchange envelopes
If both want to do so, they exchange envelopes
Should they exchange?
Suppose Berna has 20 TL. Should she exchange?
Levent Kockesen (Koc University) Strategic Form Games 31 / 1
Chairman’s Paradox
A committee of 3 members, 1, 2, 3 must choose among 3 options,A,B,C
The alternative that gets two votes is chosen
In case of a tie the chairman, player 3, chooses
Preferences1 2 3
A B CB C AC A B
Levent Kockesen (Koc University) Strategic Form Games 32 / 1
Are there dominated strategies?◮ Player 1: C is weakly dominated by A◮ Player 2: A and C are weakly dominated by B◮ Player 3: A and B are weakly dominated by C
The reduced game
2B
A 0, 1, 2B 1, 2, 0
C
Final outcome: (B,B,C) and B wins◮ Worst outcome for the chairman!
Levent Kockesen (Koc University) Strategic Form Games 34 / 1
Effort Game
You choose how much effort to expend for a joint project◮ An integer between 1 and 7
The quality of the project depends on the smallest effort: e◮ Weakest link
Effort is costly
If you choose e your payoff is
6 + 2e− e
We will play this twice
We will randomly choose one round and one student and pay her
Enter your name and effort choice for Round 1Click here for the EXCEL file
Levent Kockesen (Koc University) Strategic Form Games 35 / 1
Effort Game
Two effort level - two player version
Low HighLow 7, 7 7, 1High 1, 7 13, 13
Is there a dominated strategy?
If player 1 expects player 2 to choose Low, what is her best strategy(best response)?
If player 2 expects player 1 to choose Low what is its best response?
(Low, Low) is an outcome such that◮ Each player best responds, given what she believes the other will do◮ Their beliefs are correct
It is a Nash equilibrium
Is (Low,Low) the only Nash equilibrium?
Levent Kockesen (Koc University) Strategic Form Games 36 / 1
Nash Equilibrium
Nash Equilibrium
Nash equilibrium is a strategy profile (a collection of strategies, one foreach player) such that each strategy is a best response (maximizes payoff)to all the other strategies
Nash equilibrium is self-enforcing: no player has an incentive todeviate unilaterally
One way to find Nash equilibrium is to first find the best responsecorrespondence for each player
◮ Best response correspondence gives the set of payoff maximizingstrategies for each strategy profile of the other players
... and then find where they “intersect”
Levent Kockesen (Koc University) Strategic Form Games 37 / 1
Nash Equilibrium
Low HighLow 7, 7 7, 1High 1, 7 13, 13
1’s best response to Low is Low
Her best response to High is High
Similarly for player 2
Best response correspondences intersect at (Low, Low) and (High,High)
These two strategy profiles are the two Nash equilibria of this game
We would expect in the long-run one of these outcomes to prevail
Risk dominance
Payoff dominance
Levent Kockesen (Koc University) Strategic Form Games 38 / 1
Stag Hunt
Jean-Jacques Rousseau in A Discourse on Inequality
If it was a matter of hunting a deer, everyone well realized thathe must remain faithful to his post; but if a hare happened topass within reach of one of them, we cannot doubt that he wouldhave gone off in pursuit of it without scruple...
Stag HareStag 2, 2 0, 1Hare 1, 0 1, 1
Levent Kockesen (Koc University) Strategic Form Games 39 / 1
The Bar Scene
Levent Kockesen (Koc University) Strategic Form Games 40 / 1
The Bar Scene
Blonde BrunetteBlonde 0, 0 2, 1
Brunette 1, 2 1, 1
See S. Anderson and M. Engers: Participation Games: Market Entry,Coordination, and the Beautiful Blonde, Journal of EconomicBehavior and Organization, 2007
Levent Kockesen (Koc University) Strategic Form Games 41 / 1
John Nash
Levent Kockesen (Koc University) Strategic Form Games 42 / 1
Nash Demand Game
Each of you will be randomly matched with another student
You are trying to divide 10 TL
Each writes independently how much she wants (in multiples of 1 TL)
If two numbers add up to greater than 10 TL each gets nothing
Otherwise each gets how much she wrote
Write your name and demand on the slips
I will match two randomly
Choose one pair randomly and pay themClick here for the EXCEL file
Levent Kockesen (Koc University) Strategic Form Games 43 / 1
Some Other Commonly Used Games
Hawk-Dove
Player 1
Player 2D H
D 3, 3 1, 5H 5, 1 0, 0
Coordination
Player 1
Player 2L R
L 1, 1 0, 0R 0, 0 1, 1
Player 1
Player 2L R
L 1, 1 0, 0R 0, 0 2, 2
Levent Kockesen (Koc University) Strategic Form Games 44 / 1
Some Other Commonly Used Games
Battle of the Sexes
Player 1
Player 2B S
B 2, 1 0, 0S 0, 0 1, 2
Chicken
Jim
BuzzQuit Stay
Quit 0, 0 −1, 1Stay 1,−1 −2,−2
Levent Kockesen (Koc University) Strategic Form Games 45 / 1
Some Other Commonly Used Games
Matching Pennies
Player 1
Player 2H T
H 1,−1 −1, 1T −1, 1 1,−1
Levent Kockesen (Koc University) Strategic Form Games 46 / 1
Nash Equilibrium: Discussion
Nash equilibrium as steady-state
Nash equilibrium as self-enforcing agreement
Nash equilibrium as stable norm or institution
Levent Kockesen (Koc University) Strategic Form Games 47 / 1
Evidence from Experiments
Davis and Holt, Experimental Economics◮ In simple games with unique Nash equilibrium, that outcome has
considerable drawing power
Multiple equilibria: Importance of focal points
Interdependent preferences
Levent Kockesen (Koc University) Strategic Form Games 48 / 1
Difficulties with Nash Equilibrium
Why should players play Nash equilibrium?◮ Roger Myerson: Why not?
How would you play the following?
Player 1
Player 2A B C
A 2, 2 3, 1 0, 2B 1, 3 2, 2 3, 2C 2, 0 2, 3 2, 2
What would Myerson say?◮ If you really think the other will play C, you should play B◮ But if the other goes through this reasoning, she should play A◮ in which case, you might as well play a best response: A
Levent Kockesen (Koc University) Strategic Form Games 49 / 1
Introduction to Economic and Strategic BehaviorStrategic Form Games: Applications
Auctions also differ with respect to the valuation of the bidders
1. Private value auctions◮ each bidder knows only her own value◮ artwork, antiques, memorabilia
2. Common value auctions◮ actual value of the object is the same for everyone◮ bidders have different private information about that value◮ oil field auctions, company takeovers
Consider the above model but assume that the voters are distributedaccording to any distribution function. Then the unique Nash equilibriumis to choose the median voter’s ideal policy for both candidates.
Other Models
Models with participation costs
Models with more than two players
Models with multidimensional policy space
Models with ideological candidates: Wittman (1973) Model
What is the Nash equilibrium nB and nC?No car will go to C directly because going through B takes at most 5hrsTherefore, all the cars must go through BOnce at B no car will go directly to D because going through C takesat most 5 hrsnB = nC = 4Total travel time is 9 hours!
NR p+ v1 − v2 − e, 1− p− v1 + v2 + e p+ v1 − v2, 1− p− v1 + v2
Supporting reform is dominant for minority party◮ Reform is a common theme with challengers
Reform happens if
v1 − v2 < e
Reform is more likely if v1 − v2 is small◮ If parties are relatively equal in electoral strength◮ If majority party is much stronger, it may expect greater benefits from
patronage◮ If majority party has had disproportionate access to patronage in the
past it will have more entrenched workers and more to benefit frompatronage
Reform is also more likely if there is a strong support for reform◮ That is e is high◮ Likely to be case after scandals
Tax authority wants to prevent tax evasion with minimal cost.
Decides whether to audit or not◮ Audit always catches a cheater◮ But costs 1
The tax payer decides whether to cheat or not◮ Cheating saves 20◮ If you get caught you pay a fine of 10◮ There is also non-monetary cost equivalent to 30
An easy way to figure out dominated actions is to compare expectedpayoffsLet player 2’s mixed strategy given by q = prob(L)
L RT 1, 1 1, 0M 3, 0 0, 3B 0, 1 4, 0
u1(T, q) = 1
u1(M, q) = 3q
u1(B, q) = 4(1 − q)
0
1
2
3
4
0 14/7
12/7
q
u1(., q)
u1(T, q)
u1(M, q)
u1(B, q)An action is a never bestresponse if there is no beliefthat makes that action a bestresponseT is a never best responseAn action is a NBR iff it isstrictly dominated
Mixed strategy equilibria?◮ Only one player mixes? Not possible◮ Player 1 mixes over {T,M,B}? Not possible◮ Player 1 mixes over {M,B}? Not possible◮ Player 1 mixes over {T,B}? Let p = prob(T )
q = 1/3, 1− p = p → p = 1/2◮ Player 1 mixes over {T,M}? Let p = prob(T )
Introduction to Economic and Strategic BehaviorStrategic Form Games with Incomplete Information
Levent Kockesen
Koc University
Levent Kockesen (Koc University) Bayesian Games 1 / 1
Games with Incomplete Information
Some players have incomplete information about some components ofthe game
◮ Firm does not know rival’s cost◮ Bidder does not know valuations of other bidders in an auction
We could also say some players have private information
Levent Kockesen (Koc University) Bayesian Games 2 / 1
Games with Incomplete Information
Does incomplete information matter?
Suppose you make an offer to buy out a company
If the value of the company is V it is worth 1.5V to you
The seller accepts only if the offer is at least V
If you know V what do you offer?
You know only that V is uniformly distributed over [0, 100]. Whatshould you offer?
Enter your name and your bid
Click here for the EXCEL file
Levent Kockesen (Koc University) Bayesian Games 3 / 1
Bayesian Games
We will first look at incomplete information games where playersmove simultaneously
◮ Bayesian games
Later on we will study dynamic games of incomplete information
Levent Kockesen (Koc University) Bayesian Games 4 / 1
Bayesian Games
What is new in a Bayesian game?
Each player has a type: summarizes a player’s private information
Players’ payoffs depend on types
Incomplete information can be anything about the game◮ Payoff functions◮ Actions available to others◮ Beliefs of others; beliefs of others’ beliefs of others’...
Harsanyi (1967-1968) showed that introducing types in payoffs isadequate
Each player has beliefs about others’ types
Levent Kockesen (Koc University) Bayesian Games 5 / 1
Bayesian Equilibrium
Since payoffs depend on types different types of the same player mayplay different strategies
We assume that players maximize their expected payoffs.
Bayesian Equilibrium
A strategy profile is a Bayesian equilibrium if each type of each playermaximizes her expected payoff given other players’ strategies.
Levent Kockesen (Koc University) Bayesian Games 6 / 1
An Example: Hawk-Dove
Two animals are trying to share a prey
They can be hawkish (H) or dovish (D)
Value of the prey = 2
There is a fight if both of them choose to be hawkish
If they get into a fight they both suffer injuries◮ Cost to player 1= 2◮ Cost to player 2= c2
If only one is hawkish, he gets the entire prey
Otherwise they share
H DH −1, 1 − c2 2, 0D 0, 2 1, 1
c2 can be 0 (tough type) or 2 (weak type)◮ Pl. 2 knows◮ Pl. 1 believes it is 2 with probability q
Levent Kockesen (Koc University) Bayesian Games 7 / 1
Hawk-Dove
H DH −1,−1 2, 0D 0, 2 1, 1
Weak (q)
H DH −1, 1 2, 0D 0, 2 1, 1
Tough (1− q)
Player 1 chooses H or D
Player 2’s strategy may depend on type◮ If both do the same: pooling◮ If different types play differently: separating
Four possible types of pure strategy Bayesian equilibria◮ Separating: (W : H,T : D) or (W : D,T : H)◮ Pooling: (W : H,T : H) or (W : D,T : D)
H is strictly dominant for tough type → only two possibilities
Levent Kockesen (Koc University) Bayesian Games 8 / 1
Hawk-Dove: Separating EquilibriumH D
H −1,−1 2, 0D 0, 2 1, 1
Weak (q)
H DH −1, 1 2, 0D 0, 2 1, 1
Tough (1− q)
(W : D,T : H)
For Weak to play D, Pl. 1 must be playing HExpected payoff to H
q × 2 + (1− q)× (−1) = 3q − 1
Expected payoff to D
q × 1 + (1− q)× 0 = q
H is optimal if
3q − 1 ≥ q or q ≥ 1
2
1 plays H and 2 plays (W : D,T : H) is a Bayesian equilibrium iff q ≥ 1/2.
Levent Kockesen (Koc University) Bayesian Games 9 / 1
Hawk-Dove: Pooling Equilibrium
H DH −1,−1 2, 0D 0, 2 1, 1
Weak (q)
H DH −1, 1 2, 0D 0, 2 1, 1
Tough (1− q)
(W : H,T : H)
For Weak to play H, Pl. 1 must be playing D
Expected payoff to H is −1
Expected payoff to D is 0
So D is optimal
1 plays D and 2 plays (W : H,T : H) is a Bayesian equilibrium.
Levent Kockesen (Koc University) Bayesian Games 10 / 1
Auctions
Many economic transactions are conducted through auctions
treasury bills
foreign exchange
publicly owned companies
mineral rights
airwave spectrum rights
art work
antiques
cars
houses
government contracts
Also can be thought of as auctions
takeover battles
queues
wars of attrition
lobbying contests
Levent Kockesen (Koc University) Bayesian Games 11 / 1
Auction Formats
1. Open bid auctions1.1 ascending-bid auction
⋆ aka English auction⋆ price is raised until only one bidder remains, who wins and pays the
final price
1.2 descending-bid auction⋆ aka Dutch auction⋆ price is lowered until someone accepts, who wins the object at the
current price
2. Sealed bid auctions2.1 first price auction
⋆ highest bidder wins; pays her bid
2.2 second price auction⋆ aka Vickrey auction⋆ highest bidder wins; pays the second highest bid
Levent Kockesen (Koc University) Bayesian Games 12 / 1
Auction Formats
Auctions also differ with respect to the valuation of the bidders
1. Private value auctions◮ each bidder knows only her own value◮ artwork, antiques, memorabilia
2. Common value auctions◮ actual value of the object is the same for everyone◮ bidders have different private information about that value◮ oil field auctions, company takeovers
Levent Kockesen (Koc University) Bayesian Games 13 / 1
Independent Private Values
Each bidder knows only her own valuation
Valuations are independent across bidders
Bidders have beliefs over other bidders’ values
Risk neutral bidders◮ If the winner’s value is v and pays p, her payoff is v − p
Levent Kockesen (Koc University) Bayesian Games 14 / 1
Second Price Auctions
Only one 4G license will be sold
There are 10 groups
I generated 10 random values between 0 and 100
I will now distribute the values: Keep these and don’t show it toanyone until the end of the experiment
I will now distribute paper slips where you should enter your name,value, and bid
Highest bidder wins, pays the second highest bid
I will pay the winner her net payoff: value - priceClick here for the EXCEL file
Levent Kockesen (Koc University) Bayesian Games 15 / 1
Second Price Auctions
I. Bidding your value weakly dominates bidding higher
Suppose your value is $10 but you bid $15. Three cases:
1. There is a bid higher than $15 (e.g. $20)◮ You loose either way: no difference
2. 2nd highest bid is lower than $10 (e.g. $5)◮ You win either way and pay $5: no difference
3. 2nd highest bid is between $10 and $15 (e.g. $12)◮ You loose with $10: zero payoff◮ You win with $15: loose $2
5
10 value
12
15 bid
20
Levent Kockesen (Koc University) Bayesian Games 16 / 1
Second Price Auctions
II. Bidding your value weakly dominates bidding lower
Suppose your value is $10 but you bid $5. Three cases:
1. There is a bid higher than $10 (e.g. $12)◮ You loose either way: no difference
2. 2nd highest bid is lower than $5 (e.g. $2)◮ You win either way and pay $2: no difference
3. 2nd highest bid is between $5 and $10 (e.g. $8)◮ You loose with $5: zero payoff◮ You win with $10: earn $2
2
10 value
8
5 bid
12
Levent Kockesen (Koc University) Bayesian Games 17 / 1
English Auction
Suppose you value the item at 100 TL
What is your optimal strategy?
Stay in bidding until the price exceeds 100 TL
This is a dominant strategy
If everyone plays this strategy what happens?◮ The bidder with highest value wins◮ Pays something close to second highest value
Levent Kockesen (Koc University) Bayesian Games 18 / 1
First Price Auctions
Only one 4G license will be sold
There are 10 groups
I generated 10 random values between 0 and 100
I will now distribute the values: Keep these and don’t show it toanyone until the end of the experiment
I will now distribute paper slips where you should enter your name,value, and bid
Highest bidder wins, pays her bid
I will pay the winner her net payoff: value - priceClick here for the EXCEL file
Levent Kockesen (Koc University) Bayesian Games 19 / 1
First Price Auctions
Would you bid your value?
What happens if you bid less than your value?◮ You get a positive payoff if you win◮ But your chances of winning are smaller◮ Optimal bid reflects this tradeoff
Bidding less than your value is known as bid shading
Choose your bid b to maximize
π = (v − b) prob(win)
Probability of winning depends on your bid and others’ bids
Given what you believe about the others, it is increasing in your bid
Levent Kockesen (Koc University) Bayesian Games 20 / 1
Example
Suppose your value is 6 and the highest possible value is 10
Levent Kockesen (Koc University) Bayesian Games 30 / 1
Auction Design
Good design depends on objective◮ Revenue◮ Efficiency◮ Other
One common objective is to maximize expected revenue
In the case of private independent values with the same number ofrisk neutral bidders format does not matter
Auction design is a challenge when◮ values are correlated◮ bidders are risk averse
Other design problems◮ collusion◮ entry deterrence◮ reserve price
Levent Kockesen (Koc University) Bayesian Games 31 / 1
Auction Design
Correlated values: Ascending bid auction is better
Risk averse bidders◮ Second price auction: risk aversion does not matter◮ First price auction: higher bids
Collusion: Sealed bid auctions are better to prevent collusion
Entry deterrence: Sealed bid auctions are better to promote entry
A hybrid format, such as Anglo-Dutch Auction, could be better.
Anglo-Dutch auction has two stages:
1. Ascending bid auction until only two bidders remain
2. Two remaining bidders make offers in a first price sealed bid auction
Levent Kockesen (Koc University) Bayesian Games 32 / 1
page.1
Introduction to Economic and Strategic BehaviorExtensive Form Games
Levent Kockesen
Koc University
Levent Kockesen (Koc University) Extensive Form Games 1 / 23
page.2
Extensive Form Games
Strategic form games are used to model situations in which playerschoose strategies without knowing the strategy choices of the otherplayers
In some situations players observe other players’ moves before theymove
Removing Coins:◮ There are 21 coins◮ Two players move sequentially and remove 1, 2, or 3 coins◮ Winner is who removes the last coin(s)◮ We will determine the first mover by a coin toss◮ Volunteers?
Levent Kockesen (Koc University) Extensive Form Games 2 / 23
page.3
Extensive Form Games
Strategic form has three ingredients:◮ set of players◮ sets of actions◮ payoff functions
Extensive form games provide more information◮ order of moves◮ actions available at different points in the game◮ information available throughout the game
Easiest way to represent an extensive form game is to use a game tree
Levent Kockesen (Koc University) Extensive Form Games 3 / 23
page.4
Entry Game
Kodak is contemplating entering the instant photography market andPolaroid can either fight the entry or accommodate
K
P
Out In
F A
0, 20
−5, 0 10, 10
Levent Kockesen (Koc University) Extensive Form Games 4 / 23
Half the class will play A (proposer) and half B (responder)◮ Proposers should write how much they offer to give responders◮ I will distribute them randomly to responders
BargainingTwo individuals, A and B, are trying to share a cake of size 1If A gets x and B gets y,utilities are uA(x) and uB(y)If they do not agree, A gets utility dA and B gets dBWhat is the most likely outcome?
How should the preferences of voters be aggregated?
Two alternatives (candidates, options): Easy◮ Majority Rule
More than two alternative: Problematic◮ A popular procedure: pairwise voting◮ Condorcet method: each alternative against each of the others◮ The alternative that defeats every other in pairwise voting wins
⋆ Condorcet winner
◮ Many other methods:⋆ Borda count⋆ Majority runoff, etc.
Assume that the cost of revolution is not too high or low
When elites are in power they have a commitment problem:◮ They would like to commit to some redistribution to poor◮ But their promise is not credible◮ Once the threat of revolution passes, they will renege
Acemoglu & Robinson Hypothesis: Democracy is a commitmentdevice
◮ It transfers power from elites to citizens and solves the commitmentproblem.
They may democratize or not◮ Democratization: Citizens hold power
⋆ Choose tax rate⋆ Afterwards choose whether to revolt
◮ No Democratization: Elites hold power⋆ Promise tax rate⋆ Citizens choose whether to revolt⋆ If no revolution⋆ With probability q Elites keep promise, otherwise choose tax rate again
Introduction to Economic and Strategic BehaviorRepeated Games
Levent Kockesen
Koc University
Levent Kockesen (Koc University) Repeated Games 1 / 29
page.2
A Simple Repeated Game
Everybody is matched with somebody else
Choose C or N
1. N → You gain 1 TL2. C → Other gains 2 TL
Will repeat as long as Excel draws a number ≤ 0.6Click here for the EXCEL file
Levent Kockesen (Koc University) Repeated Games 2 / 29
page.3
Prisoners’ Dilemma
Cooperate DefectCooperate 2, 2 −1, 3
Defect 3,−1 0, 0
It is a parable for many social interactions◮ Contributing to a joint project◮ Common pool resources◮ Arms’ race
The prediction is quite bleak
But humanity seems to have solved at least some of these problems◮ Many social and economic interactions depend on trust and
cooperation
We could not have survived otherwise
Levent Kockesen (Koc University) Repeated Games 3 / 29
page.4
Institutions
Institutions may establish cooperative outcomes◮ Formal institutions:
⋆ State⋆ Law⋆ Police, etc.
◮ Informal Institutions⋆ Social norms⋆ Religion⋆ Ethics, etc.
How do these institutions arise?
What kind of institutions survive?
Levent Kockesen (Koc University) Repeated Games 4 / 29
page.5
Institutions
Institutions
Any institution must be a Nash equilibrium of the game of life.
Otherwise some actor is not acting optimally
In the long-run such non-Nash behavior will be eliminated◮ We will see later that such behavior is not evolutionary stable
Levent Kockesen (Koc University) Repeated Games 5 / 29
page.6
Cooperation
Cooperation in PD type situations is a commonly observedphenomenon
But it is not a Nash equilibrium
How did it survive as an institution?
In early societies PD type games were used to be played in smallgroups in repeated manner
Could cooperation be sustained in such environments?
Levent Kockesen (Koc University) Repeated Games 6 / 29
page.7
Repeated Interactions
In repeated games, reward-punishment mechanisms can be used
Tit-for-Tat◮ Start with cooperating◮ Then do what the other has done last period
Trigger Strategy◮ Start with cooperating◮ Cooperate as long as the other has cooperated◮ After a defection, play defect forever◮ Also known as Grim-Trigger Strategy
Both are forms of reciprocal altruism◮ It is one of the universal norms◮ Known as the Golden Rule
Observed in animal world too◮ Vampire bats, stickleback fish
Levent Kockesen (Koc University) Repeated Games 7 / 29
page.8
Golden Rule
Levent Kockesen (Koc University) Repeated Games 8 / 29
page.9
Reciprocal Altruism
Cooperate DefectCooperate 2, 2 −1, 3
Defect 3,−1 0, 0
Is Tit-for-Tat or Trigger strategy a subgame perfect equilibrium?
Suppose you play it once
Twice?
Need infinite repetition◮ But is this realistic?◮ Perhaps more realistic than finite repetition◮ Every period there is a positive probability that there will be another
period of interaction
Levent Kockesen (Koc University) Repeated Games 9 / 29
page.10
Repeated Games
Players play a stage game repeatedly over time
If there is a final period: finitely repeated game
If there is no definite end period: infinitely repeated game◮ We could think of firms having infinite lives◮ Or players do not know when the game will end but assign some
probability to the event that this period could be the last one
Levent Kockesen (Koc University) Repeated Games 10 / 29
page.11
Repeated Games
Today’s payoff of $1 is more valuable than tomorrow’s $1◮ This is known as discounting◮ Think of it as probability with which the game will be played next
period◮ ... or as the factor to calculate the present value of next period’s payoff◮ or some combination
Denote the discount factor by δ ∈ (0, 1)
In PV interpretation: if interest rate is r
δ =1
1 + r
Levent Kockesen (Koc University) Repeated Games 11 / 29
page.12
Payoffs
If starting today a player receives an infinite sequence of payoffs
Levent Kockesen (Koc University) Repeated Games 12 / 29
page.13
Equilibria of Infinitely Repeated Games
There is no end period of the game
Cannot apply backward induction type algorithm
We use One-Shot Deviation Property to check whether a strategyprofile is a subgame perfect equilibrium
One-Shot Deviation Property
A strategy profile is a SPE of a repeated game if and only if no player cangain by changing her action after any history, keeping both the strategiesof the other players and the remainder of her own strategy constant
Take an history
For each player check if she has a profitable one-shot deviation (OSD)
Do that for each possible history
If no player has a profitable OSD after any history you have an SPE
If there is at least one history after which at least one player has aprofitable OSD, the strategy profile is NOT an SPE
Levent Kockesen (Koc University) Repeated Games 13 / 29
page.14
Reciprocal Altruism
Cooperate DefectCooperate 2, 2 −1, 3
Defect 3,−1 0, 0
Consider the following trigger strategy◮ Start with Cooperate◮ Cooperate as long as everyone has always cooperated◮ Defect otherwise
Is it a subgame perfect equilibrium?
Suppose you think the other player will play according to that rule
Do you have an incentive to deviate from it?
There are two types of histories:1. Cooperative Phase
⋆ Nobody has ever defected ⇒ Cooperate
2. Punishment Phase⋆ Somebody has defected at some point ⇒ Defect
Levent Kockesen (Koc University) Repeated Games 14 / 29
page.15
Reciprocal AltruismCooperate Defect
Cooperate 2, 2 −1, 3Defect 3,−1 0, 0
Cooperative Phase
If you cooperate you expect to receive 2 forever◮ Payoff is
(1− δ)(2 + δ2 + δ22 + δ32 · · · ) = 2
If you defect◮ You get 3 today and 0 from tomorrow onwards◮ Payoff is
(1− δ)3
Cooperate is optimal if 2 ≥ (1− δ)3 or
δ ≥ 1
3
Levent Kockesen (Koc University) Repeated Games 15 / 29
page.16
Reciprocal Altruism
Cooperate DefectCooperate 2, 2 −1, 3
Defect 3,−1 0, 0
Punishment Phase
Defecting gets you 0
If you cooperate◮ You get −1 today◮ At most 0 starting tomorrows
Defect is optimal
Trigger is an equilibrium if and only if δ ≥ 1/3.
Cooperation can be sustained if players are patient enough and futureinteraction is likely enough.
Levent Kockesen (Koc University) Repeated Games 16 / 29
page.17
Reciprocal Altruism
More general payoffs
Cost of Cooperation c > 0
Benefit to the other b > 0
Cooperate DefectCooperate b− c, b− c −c, b
Defect b,−c 0, 0
Condition for equilibrium
b− c ≥ (1− δ)b
or
δ ≥ c
b
Levent Kockesen (Koc University) Repeated Games 17 / 29
page.18
Reciprocal Altruism
Reciprocal Altruism is an equilibrium norm if
1. Future interaction is likely enough
2. Cost of altruism is small enough
3. Benefit of altruism is large enough
This provides testable predictions◮ In times of instability we should see more defection◮ If short term gain is too large, cooperation is harder to sustain
Also important:◮ Must be able to identify cheaters◮ Trigger strategy is inefficient
⋆ Limited punishment may be enough
Levent Kockesen (Koc University) Repeated Games 18 / 29
page.19
Reciprocal Altruism
David Hume (1740): A Treatise of Human Nature
I learn to do service to another, without bearing him any real kindness,because I foresee, that he will return my service in expectation of anotherof the same kind, and in order to maintain the same correspondence ofgood offices with me and others. And accordingly, after I have served himand he is in possession of the advantage arising from my action, he isinduced to perform his part, as foreseeing the consequences of his refusal.
Levent Kockesen (Koc University) Repeated Games 19 / 29
page.20
Theory of Repeated Games
Other equilibria?
Many!
Folk Theorem
If the likelihood of future interaction is high enough, any individuallyrational payoff can be sustained as an equilibrium payoff.
Institutions have another role: Selecting equilibrium◮ Efficiency◮ Fairness
Societies whose institutions select efficient and fair allocations thrive,others perish.
Levent Kockesen (Koc University) Repeated Games 20 / 29
page.21
Evidence: Stickleback Fish
When a potential predator appears, one or more sticklebacksapproach to check it out
This is dangerous but provides useful information◮ If hungry predator, escape◮ Otherwise stay
Milinski (1987) found that they use Tit-for-Tat like strategy◮ Two sticklebacks swim together in short spurts toward the predator
Cooperate: Move forward
Defect: Hang back
Levent Kockesen (Koc University) Repeated Games 21 / 29
page.22
Evidence: Stickleback Fish
Milinski also run an ingenious experiment
Used a mirror to simulate a cooperating or defecting stickleback
When the mirror gave the impression of a cooperating stickleback◮ The subject stickleback move forward
When the mirror gave the impression of a defecting stickleback◮ The subject stickleback stayed back
Levent Kockesen (Koc University) Repeated Games 22 / 29
page.23
Evidence: Vampire Bats
Vampire bats (Desmodus rotundus) starve after 60 hours
They feed each other by regurgitatingIs it kin selection or reciprocal altruism?
◮ Kin selection: Costly behavior that contribute to reproductive successof relatives
Wilkinson, G.S. (1984), Reciprocal food sharing in the vampire bat,Nature.
◮ Studied them in wild and in captivation
Levent Kockesen (Koc University) Repeated Games 23 / 29
page.24
Evidence: Vampire Bats
If a bat has more than 24 hours to starvation it is usually not fed◮ Benefit of cooperation is high
Primary social unit is the female group◮ They have opportunities for reciprocity
Adult females feed their young, other young, and each other◮ Does not seem to be only kin selection
Unrelated bats often formed a buddy system, with two individualsfeeding mostly each other
◮ Reciprocity
Also those who received blood more likely to donate later on
If not in the same group, a bat is not fed◮ If not associated, reciprocation is not very likely
It is not only kin selection
Levent Kockesen (Koc University) Repeated Games 24 / 29
page.25
Evidence: Medieval Trade Fairs
In 12th and 13th century Europe long distance trade took place infairs
Transactions took place through transfer of goods in exchange ofpromissory note to be paid at the next fair
Room for cheating
No established commercial law or state enforcement of contracts
Fairs were largely self-regulated through Lex mercatoria, the”merchant law”
◮ Functioned as the international law of commerce◮ Disputes adjudicated by a local official or a private merchant◮ But they had very limited power to enforce judgments
Has been very successful and under lex mercatoria, trade flourished
How did it work?
Levent Kockesen (Koc University) Repeated Games 25 / 29
page.26
Evidence: Medieval Trade Fairs
What prevents cheating by a merchant?
Could be sanctions by other merchants
But then why do you need a legal system?
What is the role of a third party with no authority to enforcejudgments?
Levent Kockesen (Koc University) Repeated Games 26 / 29
page.27
Evidence: Medieval Trade Fairs
If two merchants interact repeatedly honesty can be sustained bytrigger strategy
In the case of trade fairs, this is not necessarily the case
Can modify trigger strategy◮ Behave honestly iff neither party has ever cheated anybody in the past
Requires information on the other merchant’s past
There lies the role of the third party
Levent Kockesen (Koc University) Repeated Games 27 / 29
page.28
Evidence: Medieval Trade Fairs
Milgrom, North, and Weingast (1990) construct a model to show howthis can work
The stage game:
1. Traders may, at a cost, query the judge, who publicly reports whetherany trader has any unpaid judgments
2. Two traders play the prisoners’ dilemma game3. If queried before, either may appeal at a cost4. If appealed, judge awards damages to the plaintiff if he has been
honest and his partner cheated5. Defendant chooses to pay or not6. Unpaid judgments are recorded by the judge
Levent Kockesen (Koc University) Repeated Games 28 / 29
page.29
Evidence: Medieval Trade Fairs
If the cost of querying and appeal are not too high and players aresufficiently patient the following strategy is a subgame perfectequilibrium:
1. A trader querries if he has no unpaid judgments2. If either fails to query or if query establishes at least one has unpaid
judgement play Cheat, otherwise play Honest3. If both queried and exactly one cheated, victim appeals4. If a valid appeal is filed, judge awards damages to victim5. Defendant pays judgement iff he has no other unpaid judgements
This supports honest trade
An excellent illustration the role of institutions◮ An institution does not need to punish bad behavior, it just needs to
help people do so
Levent Kockesen (Koc University) Repeated Games 29 / 29
page.1
Introduction to Economic and Strategic BehaviorExtensive Form Games with Incomplete Information
Levent Kockesen
Koc University
Levent Kockesen (Koc University) Ext. Form Inc. Info 1 / 38
page.2
Extensive Form Games with Incomplete Information
We have seen extensive form games with perfect information◮ Entry game
And strategic form games with incomplete information◮ Auctions
Many incomplete information games are dynamic
There is a player with private information
Levent Kockesen (Koc University) Ext. Form Inc. Info 2 / 38
page.3
Used Car Market
Two types of used cars:◮ Bad (Lemon)◮ Good (Peach)
Seller knows
Potential buyer does not◮ Believes a fraction q is peach
Willingness to pay (or be paid)
Peach LemonBuyer 16,000 6,000Seller 12,500 3,000
Levent Kockesen (Koc University) Ext. Form Inc. Info 3 / 38
page.4
Used Car Market
Peach LemonBuyer 16,000 6,000Seller 12,500 3,000
If there is complete information
Both types are brought to the market and sold◮ This is the efficient outcome
Price?
Depends on bargaining power◮ Assume seller has all the bargaining power
Lemon’s price = 6,000
Peach’s price = 16,000
Levent Kockesen (Koc University) Ext. Form Inc. Info 4 / 38
page.5
Used Car Market
Peach LemonBuyer 16,000 6,000Seller 12,500 3,000
Back to incomplete information
Is there an efficient equilibrium?
If both are brought to the market expected value to the buyers
q × 16, 000 + (1− q)× 6, 000 = 6, 000 + 10, 000q
This is the most they are willing to pay for a car
Peach owner must be willing to sell at that price:
6, 000 + 10, 000q ≥ 12, 500
or q ≥ 0.65
Levent Kockesen (Koc University) Ext. Form Inc. Info 5 / 38
page.6
Lemon’s Problem
There is an efficient equilibrium iff fraction of lemons is small
Otherwise only lemons are brought to the market
First analyzed by George Akerlof◮ Winner of 2001 Nobel in Economics◮ Akerlof, G. (1970), The Market for Lemons, Quarterly Journal of
Economics.
Levent Kockesen (Koc University) Ext. Form Inc. Info 6 / 38
page.7
Adverse Selection
This is also known as adverse selection
Comes from the insurance industry
Suppose the average risk of accident is 10%
Cost of accident is 100,000
Insurance company breaks even if premium is 10,000
But then only those whose risk is greater than 10% will buy insurance
Insurance company will make losses
Increase premium?◮ Say 15,000
What happens?
Levent Kockesen (Koc University) Ext. Form Inc. Info 7 / 38
page.8
Signaling and Screening
Two possible but usually costly solutions◮ Signal◮ Screen
Signaling Games: Informed player moves first◮ Warranties◮ Education
Screening Games: Uninformed player moves first◮ Insurance company offers different contracts◮ Price discrimination
Levent Kockesen (Koc University) Ext. Form Inc. Info 8 / 38
page.9
Signaling Examples
Used-car dealer◮ How do you signal quality of your car?◮ Issue a warranty
An MBA degree◮ How do you signal your ability to prospective employers?◮ Get an MBA
Entrepreneur seeking finance◮ You have a high return project. How do you get financed?◮ Retain some equity
Stock repurchases◮ Often result in an increase in the price of the stock◮ Manager knows the financial health of the company◮ A repurchase announcement signals that the current price is low
Limit pricing to deter entry◮ Low price signals low cost
Peacock tails
Levent Kockesen (Koc University) Ext. Form Inc. Info 9 / 38
page.10
An Example: Beer or Quiche
A stranger comes to the bar for breakfast
A native is also at the bar trying to decide whether to pick a fight
The stranger is either tough or weak◮ Native doesn’t know
Stranger’s payoff to◮ Fight is 2◮ No fight is 4
Native’s payoff to◮ Fight is 2 against weak, −1 against tough◮ No fight is 0
Stranger decides on breakfast◮ Quiche: no cost◮ Beer: costs
⋆ 1 to tough⋆ 3 to weak
Levent Kockesen (Koc University) Ext. Form Inc. Info 10 / 38
page.11
Beer or Quiche
Nature
1
1
2, 2
4, 0
2,−1
4, 0
−1, 2
1, 0
1,−1
3, 0
W〈0.5〉
T〈0.5〉
Q
Q
2
f
r
f
r
B
B
2
f
r
f
r
How would you play?
Backward induction or subgame perfect equilibrium?
Beliefs are important
We need a new solution concept
Levent Kockesen (Koc University) Ext. Form Inc. Info 11 / 38
page.12
Beer or Quiche
Nature
1
1
2, 2
4, 0
2,−1
4, 0
−1, 2
1, 0
1,−1
3, 0
W〈0.5〉
T〈0.5〉
Q
Q
2
f
r
f
r
B
B
2
f
r
f
r
Given beliefs we want players to play optimally at every informationset
◮ sequential rationality
We want beliefs to be consistent with chance moves and strategies◮ Bayes Law gives consistency
Levent Kockesen (Koc University) Ext. Form Inc. Info 12 / 38
page.13
Beer or Quiche
Nature
1
1
2, 2
4, 0
2,−1
4, 0
−1, 2
1, 0
1,−1
3, 0
W〈0.5〉
T〈0.5〉
Q
Q
2
f
r
f
r
B
B
2
f
r
f
r
If you are tough can you avoid a fight?
Drink beer?
Is this an equilibrium?
Levent Kockesen (Koc University) Ext. Form Inc. Info 13 / 38
page.14
Strategies and Beliefs
A solution in an extensive form game of incomplete information is acollection of
1. A behavioral strategy profile
2. A belief system
We call such a collection an assessment
A behavioral strategy specifies the play at each information set of theplayer
◮ This could be a pure strategy or a mixed strategy
A belief system is a probability distribution over the nodes in eachinformation set
Levent Kockesen (Koc University) Ext. Form Inc. Info 14 / 38
page.15
Perfect Bayesian Equilibrium
Sequential Rationality
At each information set, strategies must be optimal, given the beliefs andsubsequent strategies
Weak Consistency
Beliefs are determined by Bayes Law and strategies whenever possible
The qualification “whenever possible” is there because if an informationset is reached with zero probability we cannot use Bayes Law to determinebeliefs at that information set.
Perfect Bayesian Equilibrium
An assessment is a PBE if it satisfies
1. Sequentially rationality
2. Weak Consistency
Levent Kockesen (Koc University) Ext. Form Inc. Info 15 / 38
page.16
Bayes Law
Suppose you take a test for a disease
Test is 99% reliable◮ prob(+|D) = 0.99◮ prob(−|N) = 0.99
On average 0.5% of population have disease
You tested positive
What is the likelihood that you have the disease?
Levent Kockesen (Koc University) Ext. Form Inc. Info 16 / 38
Levent Kockesen (Koc University) Ext. Form Inc. Info 19 / 38
page.20
Nature
1
1
〈0〉
〈1〉
〈1〉
〈0〉
2, 2
4, 0
2,−1
4, 0
−1, 2
1, 0
1,−1
3, 0
W〈0.5〉
T〈0.5〉
Q
Q
2
f
r
f
r
B x
B
2
f
r
f
r
Separating Equilibria
2. T : Q,W : B◮ Bayes rule (BR) ⇒ B → W,Q → T◮ Sequential rationality (SR) of 2 ⇒ Q → r, B → f◮ SR of 1 ⇒ Tough: OK, Weak: NO◮ No such PBE
Levent Kockesen (Koc University) Ext. Form Inc. Info 20 / 38
page.21
Nature
1
1
〈〉
〈〉
〈0.5〉
〈0.5〉
2, 2
4, 0
2,−1
4, 0
−1, 2
1, 0
1,−1
3, 0
W〈0.5〉
T〈0.5〉
Q
Q
2
f
r
f
r
B
B
2
f
r
f
r
Pooling Equilibria
1. T : B,W : B◮ Bayes rule (BR) ⇒ prob(T |B) = 0.5, prob(T |Q) = free◮ Sequential rationality (SR) of 2 ⇒ B → f,Q →?◮ SR of 1 ⇒ Tough: NO, Weak: NO◮ No such PBE
Levent Kockesen (Koc University) Ext. Form Inc. Info 21 / 38
page.22
Nature
1
1
〈0.5〉
〈0.5〉
〈〉
〈〉
2, 2
4, 0
2,−1
4, 0
−1, 2
1, 0
1,−1
3, 0
W〈0.5〉
T〈0.5〉
Q
Q
2
f
r
f
r
B
B
2
f
r
f
r
Pooling Equilibria
2. T : Q,W : Q◮ Bayes rule (BR) ⇒ prob(T |Q) = 0.5, prob(T |B) = free◮ Sequential rationality (SR) of 2 ⇒ Q → f,B →?◮ But SR of player 1 type T ⇒ B → f◮ SR of 2 ⇒ prob(W |B) = 1
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Beer or Quiche
What if the cost of beer is equal to 1 for weak stranger too?
Nature
1
1
2, 2
4, 0
2,−1
4, 0
1, 2
3, 0
1,−1
3, 0
W〈0.5〉
T〈0.5〉
Q
Q
2
f
r
f
r
B
B
2
f
r
f
r
Can Beer signal toughness?
Signal must be costlier for the weak
Known as the handicap principle in biology
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page.25
Peacock’s Tail
Why?
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page.26
Peacock’s Tail
Could peahen be the reason?
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page.27
Handicap Principle
Why do we observe traits that reduce fitness?
Because they are signals
Handicap Principle
Reliable signals must be costly to the signaler, costing the signalersomething that could not be afforded by an individual with less of aparticular trait
First proposed by biologist Amotz Zahavi◮ Zahavi, A. (1975) Mate selection - a selection for a handicap. Journal
of Theoretical Biology.
Grafen, A. (1990) Biological signals as handicaps. Journal ofTheoretical Biology
◮ Game theoretical modeling
Levent Kockesen (Koc University) Ext. Form Inc. Info 27 / 38
page.28
Peacock’s Tail
Peacocks have different genetic quality◮ High (H) or Low (L)
Peahens’ fitness higher if they mate with high quality
Peacocks’ fitness higher if they are chosen as mates
Peacocks may exhibit different sized tails◮ Short (S) or Long (L)
Long tail is costly◮ cH to high quality◮ cL to low quality
Is there an equilibrium in which peahens select long tailed peacocks asmates?
◮ Known as sexual selection
Levent Kockesen (Koc University) Ext. Form Inc. Info 28 / 38
page.29
Peacock’s Tail
Nature
Pc
Pc
1, 1
0, 0
1, 0
0, 1
1− cH , 1
0, 0
1− cL, 0
0, 1
H〈0.5〉
L〈0.5〉
S
S
Ph
m
r
m
r
L
L
Ph
m
r
m
r
Levent Kockesen (Koc University) Ext. Form Inc. Info 29 / 38
page.30
Peacock’s Tail
Chance
Pc
Pc
1, 1
0, 0
1, 0
0, 1
1− cH , 1
0, 0
1− cL, 0
0, 1
H〈0.5〉
L〈0.5〉
S
S
Ph
m
r
m
r
L
L
Ph
m
r
m
r
Equilibrium with only High quality “choosing” Long tail?
Long must be interpreted as High quality
Short must be interpreted as Low quality
cH < 1
cL > 1
Levent Kockesen (Koc University) Ext. Form Inc. Info 30 / 38
page.31
Alternative Theories of Education
Human Capital Theory◮ Education enhances productivity and increases earnings◮ Individuals decide comparing marginal benefit and cost of additional
education
Signaling Theory◮ Higher education signals higher ability and increases earnings◮ It pays only more able to get higher education
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page.32
Signaling Theory of Education
First analyzed by Michael Spence◮ Winner of Economics Nobel in 2001◮ Spence, A. M. (1973). Job Market Signaling. Quarterly Journal of
Economics.
There are two types of workers◮ high ability (H): proportion q◮ low ability (L): proportion 1− q
Output is equal to◮ H if high ability◮ L if low ability
Workers can choose to have High education (H) or Low education (L)
Low education costs nothing but High costs◮ cH if high ability◮ cL if low ability
There are many employers bidding for workers◮ Wage of a worker is equal to her expected output
High education is completely useless in terms of worker’s productivity!
Levent Kockesen (Koc University) Ext. Form Inc. Info 32 / 38
page.33
A Theory of Education
If employers can tell worker’s ability wages will be given by
wH = H,wL = L
Nobody gets High education
Best outcome for high ability workers
If employers can only see worker’s education, wage can only depend oneducation
Employers need to form beliefs about ability in offering a wage
wH = pH ×H + (1− pH)× L
wL = pL ×H + (1− pL)× L
where pH (pL) is employers’ belief that worker is high ability if shehas High (Low) education
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page.34
Separating EquilibriaOnly High ability gets High education
What does Bayes Law imply?
pH = 1, pL = 0
What are the wages?wH = H,wL = L
What does High ability worker’s sequential rationality imply?
H − cH ≥ L
What does Low ability worker’s sequential rationality imply?
L ≥ H − cL
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page.35
Separating Equilibria
Only High ability gets High education
CombiningcH ≤ H − L ≤ cL
Again the same principle as in Beer or Quiche and Peacock’s Tail
There is an equilibrium in which only High ability gets High education iff
cH ≤ H − L ≤ cL
High education is a waste of money but High ability does it just tosignal her ability
The result would be even stronger if education increases productivity◮ This is an alternative but complementary theory of education◮ Known as human capital theory
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page.36
Evidence
Two papers
Bedard, K. (2001), “Human Capital Versus Signalling Models:University Access and High School Drop-outs,” Journal of PoliticalEconomy 109, 749-775.
Land, K. and D. Kropp (1986), “Human Capital Versus Sorting: TheEffects of Compulsory Attendance Laws,” Quarterly Journal ofEconomics, 101, 609-624.
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page.37
Bedard (2001)Tests signaling model of education against the human capital modelUses ease of access to university (presence of a local university) as aninstrumentHuman capital model:
◮ Education enhances productivity and increases earnings◮ Individuals decide comparing marginal benefit and cost of additional
education◮ Easier access to university ⇒ More high-school graduates go to
universitySignaling model
◮ Easier access to university ⇒ High ability high-school graduates go touniversity
◮ Average ability of high-school graduates becomes lower◮ Incentives to pool with them become weaker◮ Low ability students drop out of high school
The paper finds that easier university access1. increases the probability to dropout from high school2. increases the average skill of dropouts
Rejects human capital model in favor of the signaling modelLevent Kockesen (Koc University) Ext. Form Inc. Info 37 / 38
page.38
Signaling: Lang and Kropp (1986)
Tests signaling model of education against human capital model
Uses compulsory school attendance law (CAL) as an instrumentSignaling model:
◮ Suppose CAL increases minimum education from s− 1 to s years◮ Lower ability workers who would have left school after s− 1 will now
remain in school through s◮ Average ability and therefore wage of workers with s years of education
decrease◮ For the more able workers who previously stopped at s years, obtaining
s+ 1 years of education becomes more attractive, and so on◮ Prediction: CAL increases educational attainment of high-ability
individuals who are not directly affected by the law
Human capital model:◮ Prediction: No change in educational attainment of those who are not
directly affected by the law
They find CAL indeed increases the education received by those whoare not directly affected by the law and hence reject human capitalmodel in favor of the signaling model
Levent Kockesen (Koc University) Ext. Form Inc. Info 38 / 38
page.1
Introduction to Economic and Strategic BehaviorExtensive Form Games with Incomplete Information
Levent Kockesen
Koc University
Levent Kockesen (Koc University) Ext. Form Inc. Info 1 / 30
page.2
Screening
A contracting problem with Hidden Information
Uninformed party (principal) offers contract to informed party (agent)
Examples◮ Insurance
⋆ Insuree knows her risk, insurer does not⋆ Insurer offers several packages with different premiums and deductibles
◮ Finance⋆ Borrower knows the risk of project, lender does not⋆ Lender offers several packages with different interest rates and
collateral requirements
◮ Hiring⋆ Applicants know their ability, employer does not⋆ Employer offers different packages varying wages, bonuses, etc.
◮ Pricing⋆ Buyer knows her valuation of the product, seller does not⋆ Seller offers different qualities at different prices, or quantity discounts
Levent Kockesen (Koc University) Ext. Form Inc. Info 2 / 30
page.3
Screening by a Monopolist: A Very Simple Example
You are a wine producer
A fraction c of your customers are connoisseurs while the rest arenovice
You can produce high quality and/or low quality wine
The following table gives the cost of producing the two qualities ofwine as well as the willingness to pay for them by the two types ofcustomers
Willingness to payCost Connoisseur Novice
Low Quality 2 10 6High Quality 10 30 12
You choose which qualities to produce and their prices to maximizeyour expected profit
Levent Kockesen (Koc University) Ext. Form Inc. Info 3 / 30
page.4
Screening by a Monopolist
Suppose you can tell your customer’s type
What should you do?
Produce both qualities
Sell high quality to connoisseur at $30 and low quality to novice at $6
Your profit is
(30− 10)× c+ (6− 2)× (1− c) = 20c + 4(1− c)
Maximizes the total surplus:◮ Extra benefit from high quality is bigger (smaller) than the extra cost
for connoisseur (novice)
This is first degree (or perfect) price discrimination
It is great if you can implement it
But usually you cannot observe types
You may try to ask them their types. What would they tell you?
Levent Kockesen (Koc University) Ext. Form Inc. Info 4 / 30
page.5
You cannot observe types
What happens if you offer high quality at $30 and low quality at $6?
Here are the net payoffs of the two types from purchasing the twoqualities
Less than if you could observe types and perfectly discriminate
What is the best you can do in this case?
Levent Kockesen (Koc University) Ext. Form Inc. Info 5 / 30
page.6
You cannot observe types
There are several things that you can do
1. Offer only low quality wine
2. Offer only high quality wine
3. Offer both so that Connoisseur (C) chooses high and Novice (N)chooses low quality
Levent Kockesen (Koc University) Ext. Form Inc. Info 6 / 30
page.7
Offer only low quality
Willingness to payCost Connoisseur Novice
Low Quality 2 10 6High Quality 10 30 12
What price should you set?
If you set price at $6 both types buy◮ Your profit is 6− 2 = 4
If you set price at $10 only type C buys◮ Your profit is c(10− 2) = 8c
Therefore your best pricing policy and resulting profit is
Price Profitc ≤ 1/2 6 4c > 1/2 10 8c
Levent Kockesen (Koc University) Ext. Form Inc. Info 7 / 30
page.8
Offer only high quality
Willingness to payCost Connoisseur Novice
Low Quality 2 10 6High Quality 10 30 12
What price should you set?
If you set price at $12 both types buy◮ Your profit is 12− 10 = 2
If you set price at $30 only type C buys◮ Your profit is c(30− 10) = 20c
Therefore your best pricing policy and resulting profit is
Price Profitc ≤ 1/10 12 2c > 1/10 30 20c
Levent Kockesen (Koc University) Ext. Form Inc. Info 8 / 30
page.9
You offer both qualities
You want C to buy high and N to buy low quality wine
You want to choose PH and PL to maximize your expected profit
c(PH − 10) + (1− c)(PL − 2)
C must prefer high to low quality
30− PH ≥ 10− PL (ICC)
N must prefer low to high quality
6− PL ≥ 12− PH (ICN )
Called Incentive Compatibility (IC) constraints
And you want them to prefer to buy
30− PH ≥ 0 (IRC)
6− PL ≥ 0 (IRN )
Called Individual Rationality (IR) (or participation) constraints
Levent Kockesen (Koc University) Ext. Form Inc. Info 9 / 30
page.10
Solving the problem1. (IRN ) holds with equality: PL = 6
Suppose instead that 6− PL > 0. Then
30− PH ≥ 10 − PL > 6− PL > 0
◮ We could increase PH and PL by the same amount without violatingIC constraints and increase our profit
◮ This also shows that we can neglect (IRC)
2. (ICC) holds with equality: 30 − PH = 10− PL
Suppose instead that 30− PH > 10− PL. Then
30− PH > 10 − PL > 6− PL = 0
◮ We could increase PH without violating any constraints and increaseour profit
3. Therefore, we have
PL = 6, PH = PL + (30 − 10) = 30− (10− 6) = 26
Levent Kockesen (Koc University) Ext. Form Inc. Info 10 / 30
page.11
You offer both quantities
So prices are PL = 6 and PH = 26
Notice that you extract the full surplus from the Novice but leave $4surplus to the Connoisseur
Why?Remember what happens if you set prices at perfect informationlevels: i.e. PL = 6 and PH = 30
◮ Novice will happily buy his low quality wine◮ Too bad: Connoisseurs want to buy low quality as well
The amount of surplus you leave to Connoisseur is the minimum youhave to give her to buy high quality
◮ This is called information rent
What is your profit?
(26− 10)× c+ (6− 2)× (1− c) = 16c + 4(1− c)
Compare this with your profit under observable types
(30− 10)× c+ (6− 2)× (1− c) = 20c + 4(1− c)
Levent Kockesen (Koc University) Ext. Form Inc. Info 11 / 30
page.12
What is the best?
Low Only High Only BothPrice Profit Price Profit Price Profit
c ≤ 1/10 6 4 12 2 (26, 6) 16c+ 4(1− c)1/10 < c ≤ 1/2 6 4 30 20c (26, 6) 16c+ 4(1− c)
c > 1/2 10 8c 30 20c (26, 6) 16c+ 4(1− c)
Both is always better than Low only
If c ≤ 1/2 producing both is the best
If c > 1/2 producing High only is the best
Getting surplus from Novice customers is costly◮ You have to leave the Connoisseurs some surplus◮ As the fraction of Connoisseurs increases the benefit decreases and the
cost increases
Therefore, if there are enough Connoisseurs around you don’t caremuch about getting the surplus from the Novice
Levent Kockesen (Koc University) Ext. Form Inc. Info 12 / 30
page.13
Insurance
Similarly in insurance markets
Different policies with different premiums and coverage
High Premium High Coverage → High risk chooses
Low Premium Low Coverage → Low risk chooses
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page.14
Insurance
Why are there insurance firms?
They only shift risk from one party to another
Insurance could be beneficial for both the insurer and the insuree
Suppose cost of accident is 100
Probability is 10%
If individual is risk averse◮ She is willing to pay a little bit more than 10, say 12
If the insurance firm is risk neutral◮ It is willing to sell full insurance at little bit more than 10
At 11 they are both better off
Levent Kockesen (Koc University) Ext. Form Inc. Info 14 / 30
page.15
Insurance
This is a very general principle
Efficient Risk Sharing
More of the risk should be borne by less risk averse party. In particular, allthe risk should be borne by the risk neutral party.
Insurance companies are risk neutral because they hold many policieswith imperfectly correlated risks
Same as you holding a diversified portfolio
Levent Kockesen (Koc University) Ext. Form Inc. Info 15 / 30
page.16
Moral Hazard
But if all the risk is borne by the insurer, the insuree may act careless
This may increase accident probability◮ Known as moral hazard
If preventive care is unobservable, leave some risk on insuree
Moral hazard is a very pervasive phenomenon◮ Compensation◮ Development aid◮ Financial bail-outs
Levent Kockesen (Koc University) Ext. Form Inc. Info 16 / 30
page.17
Moral Hazard
There is moral hazard when
Agent takes an action that affects his payoff as well as the principal’s◮ How hard to work
Principal only observes an outcome, an imperfect indicator of theaction
◮ How many items the salesperson sold
Agent chooses an inefficient action◮ Salesperson slacks off
Levent Kockesen (Koc University) Ext. Form Inc. Info 17 / 30
page.18
Moral Hazard
Principal’s problem is to find a contract that induces high effort◮ Principal-Agent Problem
If the agent is risk neutral (and there is no limited liability) efficientoutcome can still be obtained
If he is risk averse then there is a trade off between◮ risk sharing: agent’s wage should not depend too strongly on outcome◮ incentives: to get high effort wage must depend on outcome
Levent Kockesen (Koc University) Ext. Form Inc. Info 18 / 30
page.19
Risk Aversion
An individual is risk averse is he prefers to receive the expected value of alottery to playing the lottery
Payoff function (of certain outcome) is
Linear if risk neutral◮ u(x) = x
Concave if risk averse◮ u(x) =
√x
Convex if risk lover◮ u(x) = x2
Levent Kockesen (Koc University) Ext. Form Inc. Info 19 / 30
page.20
Risk Preferences
b
b
bb
x
u(x)
x1 x2E[x]
Risk averse: u(E[x]) > E[u(x)]
b
b
b
x
u(x)
x1 x2E[x]
Risk neutral: u(E[x]) = E[u(x)]
b
b
b
b
x
u(x)
x1 x2E[x]
Risk lover: u(E[x]) < E[u(x)]
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page.21
An Example
The principal will hire a manager for a new project
Revenue is◮ 500K if successful◮ 0 if not
Probability of success depends on the manager’s effort◮ High effort: 0.9◮ Low effort: 0.1
Effort is costly for the manager◮ High effort: 250K◮ Low effort: 50K
Outside option of the manager is 0
Levent Kockesen (Koc University) Ext. Form Inc. Info 21 / 30
page.22
An Example: Risk Neutral Manager
Suppose the manager is risk neutral
If you can observe his effort level◮ Pay 250K for high and 0 for low effort◮ Manager expends high effort◮ Expected profit
0.9× 500− 250 = 200
◮ This is the first best profit for the principal
Levent Kockesen (Koc University) Ext. Form Inc. Info 22 / 30
page.23
An Example: Risk Neutral Manager
What is the optimal pay schedule if effort cannot be observed?
Constant wage?
Manager will choose low effort
Then the best wage is 50K
Expected profit0.1× 500− 50 = 0
Can the principal do better?
Levent Kockesen (Koc University) Ext. Form Inc. Info 23 / 30
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page.25
An Example: Risk Averse Manager
If you offer 25K salary plus 250K bonus if successful
Expected wage of the manager: 0
But there is a downside: With positive probability negative payoff
Outside option is a sure 0 payoff
A risk averse manager would not accept it
Principal has to increase expected wage◮ How much depends on how risk averse the manager is
To induce high effort there has to be a bonus
Profit will be less than the first best
For an example see Dixit et al. p. 558-561
Levent Kockesen (Koc University) Ext. Form Inc. Info 25 / 30
page.26
Incentives
Tying compensation to performance may itself create problems
Sometimes it is not easy to measure performance◮ How do you measure a professor’s performance◮ Student evaluations?◮ Professor may become student serving
Sometime success is multi-dimensional◮ If compensation is tied to only one, other dimensions may suffer
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page.27
Incentives
In Soviet Union incentives were tied to production volume◮ Quality suffered
AT&T used to pay according to line of programs◮ Unnecessarily long programs
1980s American football player Ken O’Brian◮ Many of his passes were being intercepted◮ His club gave a new contract
⋆ Penalty for every intercepted pass
◮ It worked: His passing percentage increases⋆ But he was passing a lot less
Levent Kockesen (Koc University) Ext. Form Inc. Info 27 / 30
page.28
Incentives
Individual commissions may lead employees sabotage each other◮ See David Mamet film Glengarry Glen Ross
If measuring performance is difficult but can be subjectively assessed◮ Efficiency wage may work
⋆ If performance is good pay high wage⋆ If bad, fire
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page.29
Incentives
If task is multidimensional whether they are complements orsubstitutes matter
Farmer may be working on both apple orchard and corn field◮ Substitutes
Farmer works on both apple orchard and beehive◮ Complements
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page.30
Incentives
If complements◮ High powered incentives
If substitues◮ Low powered incentives
It is better to collect complementary tasks together
Tasks at an airport are complements◮ It is best if different airports are managed by different companies◮ and each airport managed by a single company
London airports (Heathrow, Gatwick, Stanstead) did exactly theopposite
◮ All three belong to British Airports Authority◮ Different companies manage different “tasks” in each
Outcome is not very good
Levent Kockesen (Koc University) Ext. Form Inc. Info 30 / 30
page.1
Introduction to Economic and Strategic BehaviorEvolutionary Game Theory
Levent Kockesen
Koc University
Levent Kockesen (Koc University) Evolutionary Game Theory 1 / 34
page.2
Evolution
Evolution: Change over time in one or more inherited traits in populationsof individuals
Four mechanisms of evolutionary change
1. Mutation
2. Gene Flow
3. Genetic drift
4. Natural selection
Levent Kockesen (Koc University) Evolutionary Game Theory 2 / 34
page.3
Genetic Variation
Drift and natural selection require genetic variation top operate
Sources
1. Mutations: Changes in DNA2. Gene flow: Movement of genes from one population to another3. Sex: Genetic shuffling
Levent Kockesen (Koc University) Evolutionary Game Theory 3 / 34
page.4
Natural Selection
Darwin’s biggest idea: Main mechanism behind evolution
Fitter trait becomes more common in the population
Fitness refers to how good a particular genotype (set of genes) is atleaving offspring relative to other genotypes
◮ Survival, mate-finding, reproduction all matter
Levent Kockesen (Koc University) Evolutionary Game Theory 4 / 34
page.5
Natural SelectionMany examples:
Finches’ beaks on the Galapagos Islands◮ after droughts, the finch population has deeper, stronger beaks that let
them eat tougher seeds
Moths in 19th century Manchester
◮ Both light and dark peppered moths exists◮ Industrial revolution blackened trees◮ Being dark became advantageous in hiding from predators◮ In 50 years nearly all were black
Levent Kockesen (Koc University) Evolutionary Game Theory 5 / 34
page.6
Sexual Selection
Another important mechanism of evolution
Acts on ability to obtain or successfully copulate with a mate
Peacocks, male redback spider
Levent Kockesen (Koc University) Evolutionary Game Theory 6 / 34
page.7
Evolution
It is a fascinating subject but we don’t have the time to study itfurther
Check the web: Understanding Evolution1
One of the best books:◮ Richard Dawkins (1976): The Selfish Gene
1http://evolution.berkeley.eduLevent Kockesen (Koc University) Evolutionary Game Theory 7 / 34
page.8
Evolutionary Game Theory
Fitness of an individual sometimes depends on what others do◮ Success of defending the territory depends also on whether others
defend as well
This creates a game◮ Players: Members of a population◮ Strategies: Traits - behavior, color, etc.◮ Payoffs: Fitness
Can game theory help understand animal or plant behavior?
The answer has been an enthusiastic yes
Levent Kockesen (Koc University) Evolutionary Game Theory 8 / 34
page.9
Evolutionary Game Theory: History
C. Darwin (1871): The Descent of Man◮ Used game theory informally to explain equal sex ratio◮ Eliminated it from the 2nd ed.
R. A. Fisher (1930): The Genetic Theory of Natural Selection
Reformulated the theory:◮ Suppose less female births◮ Females have better mating prospects and leave more offspring◮ Parents who give birth to more females have more grandchildren◮ Genes for female birth spread
R. C. Lewontin (1961): Evolution and the Theory of Games◮ First explicit application
John Maynard Smith: Introduced evolutionarily stable strategy◮ 1972: Game Theory and the Evolution of Fighting◮ with Price 1973: The Logic of Animal Conflict◮ 1982: Evolution and the Theory of Games
Theory of evolution influenced social sciences too
Levent Kockesen (Koc University) Evolutionary Game Theory 9 / 34
page.10
The Model
A large population of organisms with different traits
Asexual reproduction◮ Each organism reproduces on its own◮ No gene reshuffling
Within species evolution◮ Strategies and payoffs are the same◮ Symmetric games
With high probability each member adopts its parent’s strategy
With small probability chooses another strategy (mutation)
Random matching in pairs
Payoffs represent (reproductive) fitness
Central idea: Strategies with higher fitness spread, those with lowerfitness diminish
Levent Kockesen (Koc University) Evolutionary Game Theory 10 / 34
page.11
The Model
We can look at two types of population composition
1. Monomorphic: Each individual plays the same strategy2. Polymorphic: Different individuals may play different strategies
Levent Kockesen (Koc University) Evolutionary Game Theory 11 / 34
page.12
Evolutionary Game Theory
Players: Fish defending their territory
Strategies:
1. Cooperate: Join in defending2. Defect: Do not defend
Payoffs:
C DC 2, 2 0, 3D 3, 0 1, 1
Can cooperation be evolutionarily stable?◮ A strategy is evolutionarily stable if it drives out any mutant
Levent Kockesen (Koc University) Evolutionary Game Theory 12 / 34
page.13
Prisoners’ Dilemma
Suppose everybody plays C
A small fraction ε of mutants appear◮ They play D
Large population ⇒◮ Probability of meeting C type: 1− ε◮ Probability of meeting D type: ε
What is the average payoff of a cooperating type?
(1− ε)× 2 + ε× 0 = 2(1− ε)
How about for the defecting type?
(1− ε)× 3 + ε× 1 = 3− ε
Defecting type has higher fitness for any ε
They will invade the population of cooperators
Cooperation is not evolutionarily stable
Levent Kockesen (Koc University) Evolutionary Game Theory 13 / 34
page.14
Prisoners’ Dilemma
Is D evolutionarily stable?
A small fraction ε of mutants play C
Probability of meeting D type: 1− ε
Probability of meeting C type: ε
What is the average payoff of a cooperating type?
ε× 2 + (1− ε)× 0 = 2ε
How about for the defecting type?
ε× 3 + (1− ε)× 1 = 1 + 2ε
Defecting type has higher fitness for any ε
Defecting is evolutionarily stable
Levent Kockesen (Koc University) Evolutionary Game Theory 14 / 34
page.15
Prisoners’ Dilemma
Lesson 1
A dominated strategy cannot be evolutionarily stable
Lesson 2
Evolution can be inefficient
Levent Kockesen (Koc University) Evolutionary Game Theory 15 / 34
page.16
Another Example: Hawk-Dove
H DH −1,−1 4, 0D 0, 4 2, 2
Is H ESS?
H → (1− ε)× (−1) + ε× 4 = −1 + 5ε
D → (1− ε)× 0 + ε× 2 = 2ε
A small population of D invades
So, H is not an ESS
What is wrong?
(H,H) is not a Nash equilibrium◮ There is a better response to H than H itself◮ This can invade a population of H players
⋆ Most of the time they meet H⋆ Rarely they meet each other
Levent Kockesen (Koc University) Evolutionary Game Theory 16 / 34
page.17
Another Example: Hawk-Dove
H DH −1,−1 4, 0D 0, 4 2, 2
D is not an ESS either◮ We will get back to this later◮ There maybe polymorphic equilibria
Remarkable fact
ESS vs. Nash
If a strategy s is an ESS, then (s, s) must be a Nash equilibrium
Only candidates for monomorphic (pure strategy) ESS are symmetricNash equilibria
Levent Kockesen (Koc University) Evolutionary Game Theory 17 / 34
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Another Example
Is every Nash equilibrium ESS?
H DH 1, 1 2, 0D 0, 2 2, 2
(D,D) is a NE
Is D an ESS?
D → ε× 0 + (1− ε)× 2 = 2(1 − ε)
H → ε× 1 + (1− ε)× 2 = ε+ 2(1− ε)
H does as well as against D AND does better against H
D is not an ESS
What is special about (D,D)?
Every strict NE is ESS
Levent Kockesen (Koc University) Evolutionary Game Theory 18 / 34
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Evolutionary Stability
NE vs. ESS
1. ESS ⇒ ESS
2. Strict NE ⇒ ESS
Evolutionarily Stable Strategy
A strategy s is an ESS if s yields a strictly higher expected payoff thandoes any mutant as long as the population of mutants is small enough.
Evolutionarily Stable Strategy
A strategy s is an ESS if there exists ε > 0 such that for any s′ 6= s
Levent Kockesen (Koc University) Evolutionary Game Theory 19 / 34
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Evolutionarily Stable Strategy
ESS
A strategy s is an ESS if
1. (s, s) is a Nash equilibrium
2. u(s′, s) = u(s, s) ⇒ u(s, s′) > u(s′, s′)
Intuitive
Easy to check
Levent Kockesen (Koc University) Evolutionary Game Theory 20 / 34
page.21
Evolutionarily Stable Strategy
H DH 1, 1 2, 0D 0, 2 2, 2
H is an ESS
1. It is a Nash equilibrium2. No other best response
D is not an ESS
1. It is a NE2. H is another best response and against H , D does not do strictly
better than H
Levent Kockesen (Koc University) Evolutionary Game Theory 21 / 34
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Polymorphic Populations
Hawk-Dove again
H DH −1,−1 4, 0D 0, 4 2, 2
There is no (pure strategy) ESS
Could there be a mixed strategy monomorphic ESS?◮ Each organism plays randomly◮ Has to be a mixed strategy Nash equilibrium◮ Not sufficient
Another way of thinking about it: Polymorphic population◮ Is there a stable population composed of both Hawks and Doves?
Levent Kockesen (Koc University) Evolutionary Game Theory 22 / 34
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Polymorphic Populations
H DH −1,−1 4, 0D 0, 4 2, 2
Suppose a fraction p is H
Average payoffs:
H → p× (−1) + (1− p)× 4 = 4− 5p
D → p× 0 + (1− p)× 2 = 2− 2p
4− 5p > 2− 2p ⇒ H ր4− 5p < 2− 2p ⇒ D ր
Stable only if
4− 5p = 2− 2p or p = 2/3
But this is nothing other than the mixed strategy equilibrium
Levent Kockesen (Koc University) Evolutionary Game Theory 23 / 34
page.24
Polymorphic Populations
How do we know it is ESS?
A phase diagram helps
p1
4
2
−1
Fitness
2/3
We are not going to do dynamics but intuitively◮ Any payoff monotonic dynamic leads to p = 2/3
This is the unique stable population composition
Polymorphic
Levent Kockesen (Koc University) Evolutionary Game Theory 24 / 34
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Polymorphic Populations
H DH −1,−1 4, 0D 0, 4 2, 2
How can we check if p = 2/3 an ESS?
Definitions is as before
Is it a NE? Yes
Are there other best responses? Yes
Does p = 2/3 do strictly better against them than they do againstthemselves? Yes
You can calculate it. Intuitively:◮ If mutant plays p′ > 2/3, they are hurt when they face each other◮ If mutant plays p′ < 2/3, p = 2/3 strategy takes advantage
Levent Kockesen (Koc University) Evolutionary Game Theory 25 / 34
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Hawk-Dove: General Payoffs
H DH (v − c)/2, (v − c)/2 v, 0D 0, v v/2, v/2
What does the incidence of fight depend on?
Is D an ESS?◮ No: (D,D) is not NE
How about H?
Must have v ≥ c
Is H a better response to D than D itself?
u(H,D) = v > v/2 = u(D,D)
So, a population of all Hawks is stable iff v ≥ c
Whenever the prize of the contest is at least as high as the cost ofresulting fight we should observe fights
Levent Kockesen (Koc University) Evolutionary Game Theory 26 / 34
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Hawk-Dove: General Payoffs
H DH (v − c)/2, (v − c)/2 v, 0D 0, v v/2, v/2
What if v < c
Must have polymorphic population
pv − c
2+ (1− p)v = (1− p)
v
2⇒ p =
v
c
Incidence of fighting increases in prize, decreases in cost
Average payoff of the population
(1− p)v
2= (1− v
c)v
2
Payoff increases in cost of fighting
Levent Kockesen (Koc University) Evolutionary Game Theory 27 / 34
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Rock, Paper, Scissors
R P SR a, a −1, 1 1,−1P 1,−1 a, a −1, 1S −1, 1 1,−1 a, a
In RPS a = 0 but we will allow a > 0
Unique Nash Equilibrium (1/3, 1/3, 1/3)
Levent Kockesen (Koc University) Evolutionary Game Theory 28 / 34
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Rock, Scissors, Paper
R P SR a, a −1, 1 1,−1P 1,−1 a, a −1, 1S −1, 1 1,−1 a, a
Let a > 0
Is p = (1/3, 1/3, 1/3) an ESS?
Consider S
S against S yields a
p against S yields a/3
So, p is not an ESS
If the game is like RPS, we should not find a stable population
Levent Kockesen (Koc University) Evolutionary Game Theory 29 / 34
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Side-Blotched Lizards
Common Side-blotched Lizard (Uta stansburiana) common in PacificCoast of North AmericaEach males follows a fixed heritable mating strategy1. Orange-throated: Strongest, do not form strong pair bonds2. Blue-throated: Middle-sized, form strong pair bonds3. Yellow-throated: Smallest, coloration mimics females
Orange beats Blue, Blue beats Yellow, and Yellow beats OrangeSinervo and Lively (1996):
◮ 1990-91: Blue dominated◮ 1992: Orange◮ 1993-94: Yellow
Evolutionary game theory seems to be more than a theoreticalcuriosity
Levent Kockesen (Koc University) Evolutionary Game Theory 30 / 34
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Asymmetric Games
Games played between different species◮ Insects and plants◮ Predator-Prey
Games played by asymmetrically positioned members of the samespecies
◮ Current owner of a nesting ground, food, or mates versus intruder◮ Different sized animals, etc.
Many confrontations are resolved peacefully◮ Current owner of food source keeps it
Levent Kockesen (Koc University) Evolutionary Game Theory 31 / 34
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Asymmetric Games
Owner
IntruderH D
H (V − c)/2, (v − c)/2 V, 0D 0, v V/2, v/2
An individual may come to play either role in different times
A strategy conditions action on the role◮ For example: Hawk if Owner, Dove if Intruder
What is ESS?
Result
A conditional strategy is an ESS iff it is a strict Nash equilibrium.
V < c and v < c ⇒ only two strict NE◮ (H,D): Known as bourgeois strategy◮ (D,H): Known as paradoxical strategy
Levent Kockesen (Koc University) Evolutionary Game Theory 32 / 34
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Asymmetric Hawk-DoveFraction of Hawks in Owner: pFraction of Hawks in Intruder: q
q < V/c ⇒ p րp < v/c ⇒ q ր
Stable points: (1, 0) and (0, 1)
p1v/c
q
1
V/c
1
Levent Kockesen (Koc University) Evolutionary Game Theory 33 / 34
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Evidence
A male Hamadryas baboon forms a long-lasting relationship withseveral females
Kummer et al. (1974) experiment
Male A is allowed to interact with the female
Male B is in a cage and watches
When released, B does not challenge A
The same males change roles over a different female
The same outcome
Escalated fights if each perceives himself as the owner
Levent Kockesen (Koc University) Evolutionary Game Theory 34 / 34