PLAYING GAMES

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PLAYING GAMES

• Topic 2

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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.

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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?

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SOCIAL COORDINATION GAME (cont.)
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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.

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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 others 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

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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?)

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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 P1s proposal.
• If P1 accepts the proposal, the 100 is divided
  accordingly.
• If P2 rejects the proposal, neither player gets
  anything.
• If youre P1, what do you offer?
• If youre P2, what do you accept?

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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.

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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?

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SOCIAL DILEMMA GAME (cont.)
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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.

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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?

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Common Pool Resources

• Garrett Hardin, The Tragedy of the Commons,
  Science, 1968
• common pasture land vs. enclosure/common pool
  resources

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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

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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.