Oct 10, 2015
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GAME THEORY:
INSIDE OLIGOPOLY
Dr. Gong Jie
National University of Singapore
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Course Road Map
Managerial Economics
Determination ofPrices
Introduction
Demand and Supply
Consumer Theory
Production and Cost Theory
Market Structure &Profit-MaximizingPricing Decisions
Competitive Markets
Market Power & Monopoly
Pricing with Market Power
Game Theory&
Oligopoly Markets
Game Theory Fundamentals
Simultaneous-Move Game
Sequential-Move Game
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Oligopoly and Strategic Thinking
Two extremes of market structure competitive market
monopoly
Oligopoly: the market structure between the extremes A small number of sellers
Each of them secures a considerably large marketshare.
The behavior of each seller has a strong influence onothers' behavior.
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Oligopoly and Strategic Thinking
How to manage an oligopolistic firm: StrategicThinking! Strategic thinking:put yourself in others' shoes
You must figure out the action and the intention of others
when you take your action, while others think in the sameway.
The outcome depends on how you and your opponentinteract with each other.
Strategic Decision Making If I believe that my competitors are rational and act to
maximize their own profits, how should I take theirbehavior into account when making my own profit-maximizing decisions?
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Game and Game Theory
Game is any situation in whichplayers (the
participants) make strategic decisions.
Firms compete with each other by setting prices, spending
on advertising, R&D, merger & acquisition, etc. Group of consumers bid against each other in an auction.
Many continentals believe life is a game; the English think
cricket is a game. (George Mikes, Hungarian humorist)
Game Theory Game theory systemizes the strategic thinking.
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Agenda
Oligopoly and strategic thinking
Game theory as the tool for strategic thinking
Setup of a game
Nash Equilibrium
Solve for the equilibrium
Games with multiple equilibria
Mixed strategy
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Grade Game
Without showing your neighbor what you are doing, write
down on a form either the letter or the letter . Think of this
as a grade bid. We will randomly pair your form with one
other form. Neither you nor your pair will ever know with
whom you were paired.
Grades may be assigned by the following rule:
If you put and your pair puts , then you will get A and your
pair grade C.
If both put , then both will get B-.
If you put and your pair puts , then you will get C and your
pair A.
If both put , then you will both get B+.
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How to describe the Grade Game?
My grades Pairs grades
me
pair
B- A
B+C
me
pair
B- C
B+A
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What strategy should a rational person
choose in the Grade Game?
Outcome Matrix Payoff Matrix
me
pair
B-, B- A, C
B+, B+C, A
me
pair
0, 0 3, -1
1, 1-1, 3
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Lessons
To figure out what actions you should choose, a good
first step is to figure out what are your payoffs (what
do you care about) and what the other players
payoffs.You should never play a strictly dominated strategy.
Rational play by rational players can lead to bad
outcomes.
NUS students are evil.
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How to specify a game?
Who are the players?
What are the possible actionsthese players can take?
What are the payoffs associated
with each possible outcome?
Who are the participants in the market?
What is the set of potential entrants?
What is the set of bidders?
Enter; launch; merge (discrete) R&D expenditure; capacity/price
setting; production level; advertising
spending (continuous)
What are everyones profits given theprices charged, capacity installed,
products launched, advertising spent,
number of entrants, etc.?
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Key Concepts in Games
Actions and Strategies
Strategy is the decision rule that describes the actions a
player will take at each decision point.
Strategy is not a single action, but a plan of actions.
Best responses The strategy of one player that results in the best payoffs
to him/her, given the combination of other players
strategies.
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What is the likely outcome of a game?
Game theory predicts optimal strategy for each player. The optimal strategy maximizes a players payoffs given others
strategic plays.
Game theory predicts how the game is going to be played in
obviously reasonable ways.
Solution concept: Nash Equilibrium Nash Equilibrium is a strategy profile (a combination of
strategy), where no player in the game has the incentive todeviate.
In a Nash Equilibrium, a player is unable to do strictly better byunilaterally switching his/her strategy.
In a Nash Equilibrium, each player's strategy is the best responseto other players' strategies.
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How to classify a game?
The sequence of moves: sequential or simultaneous?
Are the players' interests in total conflicts or is there
any commonality?
Is the game played once or repeatedly?
Do the players have full or equal information?
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One-Shot, Simultaneous-Move Games
We use normal-form (strategic form) to represent thegame.
Players simultaneously decide their strategies.
A representation of a game indicating the players, theirpossible strategies, and the payoffs resulting fromalternative strategies.
One player chooses strategy from the row, while theother player chooses strategy from the column.
In each cell, the first entry indicates the payoff of the rowplayer, while the second entry indicates the payoff of thecolumn player.
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Prisoners Dilemma
Two prisoners are interrogated separately
Confess Dont Confess
Confess
Dont
Confess
Prisoner 2
-5, -5 0, -10
-1, -1-10, 0
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What would be the outcome (equilibrium)
of such a game?
How to solve the game? How to find the Nash
equilibrium?
Step 1:
Find one players best response to each of the possiblestrategiesplayed by the other.
Circle the payoffs of this player that result from his/her best
response and the given play of the other.
Step 2: Repeat this procedure to the other player.The combination of strategies that result in two
circles in one cell is a Nash-equilibrium.
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Prisoners Dilemma: Solve the Game
Confess Dont Confess
Confess
Dont
Confess
Prisoner 2
-5, -5 0, -10
-1, -1-10, 0
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The Nash-equilibrium of the game is given by a
strategy profile (confess, confess). Both prisoner will choose to confess.
Does this strategy profile maximize their collective
payoffs?
It doesnt. If they both deny, they can end up with -2 in total.
(dont confess, dont confess) is not an equilibrium.
These two prisoners have conflicting interests!
When economic agents have conflicting interests,
individual decision making without enforcement
cannot reach the collective optimum.
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Airlines Dilemma: the blessing from
terrorists!
Baggage policy is a nagging problem for airlines.
Passengers who carry multiple bags onto a plane slow
down the boarding process.
Why didnt airlines enforce tighter limits and force theirextra baggage?
Lets try to analyze this in a normal-form game.
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Airlines Dilemma
Generous Policy Tighter Policy
Generous
Policy
Tighter
Policy
Air l ine B
0, 0 5, -5
2, 2-5, 5
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Given the other airline adopting lenient policy, no
single airline would like to put tighter limit on carry-on baggage.
If one airline adopts tighter policy, the other airline
will benefit from a lenient policy.
Individual behavior does not make the most desirableoutcome.
After 911, the U.S. government started to enforce
carry-on baggage screening regulation.
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Dominant Strategies and Dominated
Strategy
Dominant Strategy is one that is optimal no matter
what an opponent does.
Dominated strategy is one that results in worse
payoffs than other strategies regardless of otherplayers strategic plays.
An example
A and B sell competing products and they are deciding
whether to undertake advertising campaigns.
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Payoff Matrix for Advertising Game
Advertise
Dont
Advertise
Advertise
DontAdvertise
Firm B
10, 5 15, 0
10, 26, 8
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Payoff Matrix for Advertising Game
Observations
A: regardless of B,
advertising is the best.
B: regardless of A,
advertising is the best.
Firm A
Advertise
Dont
Advertise
Advertise
Dont
Advertise
Firm B
10, 5 15, 0
10, 26, 8
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Payoff Matrix for Advertising Game
Observations
Dominant strategy for A
and B is to advertise.
Do not worry about theother player.
Equilibrium in
dominant strategyFirm A
Advertise
Dont
Advertise
Advertise
Dont
Advertise
Firm B
10, 5 15, 0
10, 26, 8
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Dominant Strategies and Dominated
Strategies
If you have a dominant strategy, play it! If you have a
dominated strategy, forget about it!
Equilibrium in dominant strategies
Outcome of a game in which each firm is doing the best itcan regardless of what its competitors are doing
Optimal strategy is determined without worrying about the
actions of other players
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Simplify the game by successive
elimination of dominated strategies
Player II
Player I U
M
B
L N R
2,1 1,-2 -4,0
0,1 0,-1 0,2
-1,4 -3,5 -2,0
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In this game, no matter what player II does, the
strategy B results in strictly worse payoffs than M
for player I. Then delete it!
Player II
Player I U
M
B
L N R
2,1 1,-2 -4,0
0,1 0,-1 0,2
-1,4 -3,5 -2,0
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In the revised game, no matter what player I does,
the strategy N is strictly dominated by both L and R.
Then delete it!
Player II
Player I U
M
B
L N R
2,1 1,-2 -4,0
0,1 0,-1 0,2
-1,4 -3,5 -2,0
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Thus, we end up with a two-by-two game.
Not every game has a dominant strategy for each
player.
The existence of a dominated strategy doesnt imply
there is a single (pure) dominant strategy.
With just two strategies for each player, if one strategy isdominant then the other must be dominated.
With more than two strategies available to each player, a
player might have dominated strategies but no dominant
strategy.
A game may not involve dominated strategy.
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Systematic Procedure for Identifying The
Nash Equilibria
Step 0: identify dominant or dominated strategies and
simplify the game.
Step 1: identify player 1s best responses to each of
player 2s strategies.Step 2: identify player 2s best responses to each of
player 1s strategies.
Step 3: see where those best responses occur together.
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Multiple Equilibria: Two drivers on the
road (in opposite directions)
Left Right
Left
Right
dr iver B
1, 1 -1, -1
1, 1-1, -1
Coordination game: Both drivers want to settle on the same choice
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Why Is It Called Coordination Game?
The best choice depends on what each player thinks
the other party is likely to do.
If the two players communicate with each other
before they take every action, they will follow whatthey agree with when they take their action,
because
They have common interest!
Their agreement is a Nash equilibrium.
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Player 1
ARTS BIZ
ARTS
BIZ
Player 2
10, 10 -5,-8
8,8-8,-5
Coordination Game2: Meeting at Canteen
You call your groupmate
when you walk out of YIH.
You learn that your
groupmate is at Central
Library.
You two decide to discuss
homework at Canteen.
However, your cell phone
battery runs out of power.
You two havent agreed at
which Canteen you will meet.
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Player 1
ARTS BIZ
ARTS
BIZ
Player 2
10, 10 -5,-8
8,8-8,-5
Pure (common interest) Coordination
Which canteen will you head for?
How many Nash-equilibria arethere in the game?
Which NE will be picked?
(ARTS, ARTS) is more likely tobe played than (BIZ, BIZ). Thisis called a focal point.
A focal point may stem from
custom, common sense,tradition, etc.
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Matching Pennies
RULES Player A and B each has a
coin and must secretly turn
the coin to heads or tails.
Then both players reveal thatchoice simultaneously.
If the coins match, player A
keeps both coins. If the coins
dont match, player B keeps
both coins.
Player A
Heads Tails
Heads
Tails
Player B
1, -1 -1, 1
1, -1-1, 1
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Matching Pennies
Pure strategy: No
Nash equilibrium
No combination of
head and tails leaves
both players better off.
Player A
Heads Tails
Heads
Tails
Player B
1, -1 -1, 1
1, -1-1, 1
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Matching Pennies: How about randomized
actions?
Player A might flip coin playing heads with probability and tails with probability.
If both players follow this strategy, there is a Nashequilibriumboth players will be doing the best theycan given what their opponent is doing.
Although the outcome is random, the expected payoffis 0 for each player.
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Lets Check Whether (0.5, 0.5) is Nash
Equilibrium
Given that player 1 randomizes between head and tailwith the probabilities 0.5 and 0.5, if player 2 plays headwith a probability p, and tail with a probability 1-p, then
player 2 ends up with
(head, head) with probability 0.5*p, payoff=-1 (head, tails) with probability 0.5*(1-p), payoff=1
(tails, head) with probability 0.5*p, payoff=1
(tails, tails) with probability 0.5*(1-p), payoff=-1
Play 2s expected payoff with this mixed strategy is:
0.5*p*(-1)+0.5*(1-p)*1+0.5*p*1+0.5*(1-p)*(-1)=0
Player 2 has no incentive to deviate fromrandomizing with prob 0.5 and 0.5, since there is noother strategy with str ictly higherpayoff.
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Pure Strategy and Mixed Strategy
If a strategy involves a single action at each single
contingency, such a strategy is a pure strategy.
Player makes a specific choice or takes a specific action.
If a Nash equilibrium involves all players playingpure strategies, the equilibrium is called Pure Strategy
Equilibrium.
Sometimes a pure-strategy equilibrium doesnt exist.
When allowing for mixed strategies, every game has
a Nash equilibrium.
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Mixed Strategy
Mixed strategies: unpredictability can have strategicvalue.
Sports: Soccer, tennis, baseball
Pricing Strategy: randomized pricing as mixed strategy
A strategy of constantly changing prices:
decreases consumers incentive to shop around as they cannot learn
from experience which firm charges the lowest price
reduces the ability of rival firms to undercut a firms prices
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Is O. Henry Wrong?
In O. Henrys novel The Gift of Magi, Della and Jim are
the young couples who are poor but love each other and
are ready to sacrifice anything for each other.
They both wish to give the other a surprise Christmas gift.Della considers selling her hair to buy a chain for Jims
watch.
Jim considers selling his watch to buy a comb for Della.
In the novel, they both do that after struggling.
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Della
Sell Watch Keep Watch
Keep Hair
Sell Hair
J im
1, 2 0,0
2,10,0
Two pure strategyequilibria: (keep hair, sell
watch) and (sell hair, keepwatch)
But there still exists amixed strategy equilibrium:Della (Jim) sells hair
(watch) with theprobability 2/3.
Surprise is costly! Theoutcome of the noveloccurs with the probability
of 2/3*2/3=4/9, more thaneither of the two bestoutcomes (keep hair, sellwatch), (sell hair, keepwatch), both with prob 2/9!
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Takeaways
Game theory is useful when your payoff depend on
choices of the other parties.
It is important to think about how other players will
play, not how you think they ought toplay.Nash-equilibrium does not necessarily correspond to
the outcome that maximizes the aggregate profit of
the players.