PLAYING GAMES Topic #2. THE SOCIAL COORDINATION GAME You are in a group of six people, each of whom has an initial holding of $50 (just enough to guarantee.

PLAYING GAMES

Topic #2

THE SOCIAL COORDINATION GAME• You are in a group of six people, each of whom has an initial holding

of $50 (just enough to guarantee that no one ends up with a net loss, regardless of the outcome of the game).

• You have the opportunity, based on your own actions and those of the others in the group, to earn an additional amount or to lose some or all of your initial holding.

• You (and each other member of the group) must choose between two actions, designated LEFT and RIGHT (no political connotations intended), which have these consequences:– (A) If you choose LEFT, you earn $10 for each other member of the

group who chooses LEFT but you lose $10 for each other member of the group who chooses RIGHT.

– (B) If you choose RIGHT, you earn $10 for each other member of the group who chooses RIGHT but you lose $10 for each other member of the group who chooses LEFT.

• In general, you earn more (or lose less) to the extent that others make the same choice you do.

SOCIAL COORDINATION GAME (cont.)

• Your goal is to maximize your own earnings, and you know that everyone else is similarly motivated.

• Version 1. Each player must make his or her choice in isolation, without talking to other players.

• Version 2. Players can talk among themselves (make deals or whatever) prior to making their choices.

• But, in both versions, final choices are made by “secret ballot,” i.e., it is a “simultaneous move” game.

Do you choose LEFT or RIGHT?

SOCIAL COORDINATION GAME (cont.)

Coordination Games• Coordination Games are non-trivial only if they are simultaneous

choice games of imperfect information.

• They are perhaps “solved” by “[Shelling] focal points.”

• They are certainly “solved” by– sequential moves with perfect information– prior communication

• and there is no incentive for deception, no reason to break a promise– repeated play

• follow majority choice in previous play is “focal point”• but problem with n = 2

– convention (drive on right/left)

• Two-player coordination with conflicting interests:– neither player wins anything if the fail to cooperate;– If they coordinate on LEFT, P1 wins $200 and P1 wins $100;– If they coordinate on RIGHT, P1 wins $100 and P2 wins $200.

Fair Division• Children know how to divide a piece of cake “fair and square”:

– One kid cuts, the other chooses.– The game has perfect information.– Both kids know each other’s preferences:

• they are both greedy cake maximizers.– So the cutting kid knows the choosing kid will take the larger piece.– “Look ahead and Reason Back” [Dixit and Nalebuff, Rule 1 (p. 34)– So the cutting kid follows the maximin principle, i.e.,

• the cutting kid aims to make the smaller piece as big as possible (i.e., make the two pieces as equal as possible).

• Pre-play communication makes no difference in Fair Division– This a zero-sum (or constant-sum) game.

• If we turn it into a simultaneous move game (without perfect information), it character changes considerably, – though maximin is still appealing for the cutter.

• The n-player generalization• Fair Division by Tacit Coordination

Fair Division by Tacit Coordination• “The Bank” puts up a sum of money ($100) which two players can share

if they can tacitly coordinate on how to divide it.• This is a simultaneous move game but it is not constant-sum.• Each player writes down (on a “secret ballot”) his requested share.

– If the sum of the two requests does not exceed $100, each player get his requested share (and the bank keeps any residual share).

– If the sum of the two requests exceeds $100, neither player gets anything.

• Discrete choice variant: can choose only 100-0, 80-20, 60-40, 40-60, 20-80, 0-100 splits

• If the players can communicate in advance, this turns into a bargaining game– side payments (with enforceable agreements?)

The Ultimatum Game• “The Bank” puts up a sum of money ($100), which two

players can share (or not).• P1 makes a proposal to P2 as to how to divide the $100

between them.• P2 accepts or rejects P1’s proposal.• If P1 accepts the proposal, the $100 is divided

accordingly.• If P2 rejects the proposal, neither player gets anything.

• If you’re P1, what do you offer?• If you’re P2, what do you accept?

THE SOCIAL DILEMMA GAME• You are in a group of five people, each of whom has an initial

holding of $15 (just enough to guarantee to no one ends up with a net loss, regardless of the outcome of the game). You have the opportunity, based on your own actions and those of the others in the group, to earn an additional amount or to lose some or all of your initial holding.

• You (and each other member of the group) must choose between two actions, designated LEFT and RIGHT (no political connotations intended), which have these consequences:

– (A) If you choose LEFT, you earn $25 and your action has no effect on any other group member;

– (B) If you choose RIGHT, you earn $50 but your action also imposes a cost of $10 on each member of the group (yourself included, so you net $40).

• So, holding constant the choices of all others, you earn $25 more by choosing RIGHT rather than LEFT.

SOCIAL DILEMMA GAME (cont.)• Your goal is to maximize your own earnings, and you know that

everyone else is similarly motivated.

• Version 1. Each player must make his or her choice in isolation, without talking to other players.

• Version 2. Players can talk among themselves (and make deals or whatever) prior to making their choices.

• But, in both versions, final choices are made by "secret ballot.”

Do you choose LEFT or RIGHT?

SOCIAL DILEMMA GAME (cont.)

Dilemma Games

• Dilemma Games are not solved by sequential moves.– Regardless of what you see other people doing, you

are but off choosing RIGHT (or ‘‘defect.”– With prior communication, probably everything would

promise to chose LEFT (or “Cooperate’), but everyone has an incentive to break this promise.

– What is needed to “solve” the game is an binding agreement (or “enforceable contract” to cooperate.

– However, repeated play may lead to more cooperation, even in the absence of a binding agreement.

Dilemma Games (cont.)

• Cars and trucks used to emit large quantities of pollutants, resulting in air pollution that was both unpleasant and unhealthful.

• In the 1970s, it became possible to reduce such pollution greatly by installing fairly inexpensive pollution-control devices on car and truck engines.

• Suppose in fact that everyone prefers – (a) the state of affairs in which everyone pays for and installs the

devices and the air is clean to – (b) the state of affairs in which no one pays for and installs the

devices and the air is polluted.

• Would (almost) everyone voluntarily install the devices? • Would a law requiring everyone to install the devices pass in

a referendum?

Common Pool ResourcesGarrett Hardin, “The Tragedy of the Commons,” Science, 1968

common pasture land vs. enclosure/common pool resources

The Centipede Game

• Initially there are two piles of money $$$ and $.• P1 can either:

– (a) take the $$$ pile and give the $ pile to P2, or– (b) take neither pile.

• If P1 chooses (a), the game is over .• If P1 chooses (b), the piles are augmented to $$$$ and $$,

and P2 chooses between (a) and (b).• The game continues until a player chooses (a) or until 100

(hence centipede) rounds (or some other fixed number) have been played

• Implications of “look ahead and reason back” (the “subgame perfect equilibrium”) as in Fair Division Game

The Dollar Auction Game• The Bank tells P1 and P2 that one of them will win a prize of

$100. • Each of P1 and P2 alternately pays $5 or more to the Bank

until one player decides to stop paying and to pull out of the competition, at which point the other player wins the $100 prize.

• This can be thought of as auctioning off the prize to the highest bidder (hence the name of the game), – with the twist that both bidders must pay their final offers, though

only the player who made the higher final offer gets the prize. • How will this game end up?

• What would happen if the players can make an enforceable agreement before the bidding starts?• So the Bank wants to make sure that P1 and P2 cannot enter

into an enforceable agreement before the bidding starts.