Bayesian Games Matthew H. Henry November 10, 2004

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Bayesian GamesMatthew H. HenryNovember 10, 2004

• References
• Axlerod, Robert. 1987. The evolution of
  strategies in iterated prisoners dilemma.
  Genetic Algorithms and Simulated Annealing. (ed.
  D. Davis) London Pitman, pp. 32-43.
• Gibbons, Robert. 1992. Game Theory for Applied
  Economists. Princeton, New Jersey Princeton
  University Press.
• Harsanyi, John C. 1967. Games with Incomplete
  Information Played by Bayesian Players, Parts I,
  II and III. Management Science 14159-182,
  320-334, 486-502.
• Sigmund, Karl. 1993. Games of Life
  Explorations in Ecology, Evolution, and
  Behaviour. Oxford, England Oxford University
  Press.

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Outline

• Static Games with Bayesian Players
• Example Scalping Tickets
• Nash Equilibria for Matrix Games with Incomplete
  Information Generals
• Nash Equilibria for Games with Asymmetric
  Information Cournot Model
• Nash Equilibria for Games with Continuous Type
  Space Auction
• Dynamic Games with Bayesian Players
• Perfect Bayesian Equilibrium for Games with
  Incomplete or Imperfect Information
• Example 3-Player Game Tree
• Signaling Games
• Perfect Bayesian Equilibrium for Signaling Games
• Example Job Market Signaling

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Static Games with Incomplete Information

• Static games
• Players move simultaneously
• No observation of opponent move history
• Games with incomplete information
• One or more players lacks full information
  regarding the payoff functions and strategies
  available
• We shall limit the information deficit to the
  player state (or type) knowledge
• Player type implies (and is implied by) payoff
  function
• Matrix games will have a unique payoff matrix for
  each player type match-up

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Example Scalping Tickets

• For example, consider a scenario in which you and
  the Cavalier are each scalping tickets for beer
  money before the UVa-Miami football game
• For every discrete round of the game, each player
  assumes one of two types and can take one of two
  actions (stand in one of two locations)
• Types Buyer or Seller
• Locations in front of Durty Nellies Pub or at
  the Frys Spring Garage
• You know that you are either buying or selling
  and you know with probability p that the Cavalier
  is buying, and selling otherwise
• Four payoff matrices for the four possible type
  match-ups
• Choose a spot to maximize profit (Durty Nellies
  or Frys Spring Garage) based on your type and
  your best guess of the Cavs type

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A Better Example from Harsanyi

• Consider two Generals A and B
• A seeks to maximize (maxmin) payoff and B seeks
  to minimize (minmax) payoff
• Fixed action profiles (a1, a2) and (b1, b2)
• Each leads an army which assumes one of two
  states Strong or Weak
• This yields four possible match-ups (AS, BS),
  (AS, BW), (AW, BS), (AW, BW) with
  corresponding payoff matrices, each having its
  own Nash equilibrium

Harsanyi
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Bayesian Players

• Each player knows his own state and estimates his
  opponents state
• Each player has a pure strategy for every
  possible match-up
• Each player forms a strategy based on the
  expected payoff
• To continue the example given by Harsanyi,
  consider the following probabilities of
  occurrence for the four possible match-ups

BS
BW
AS
4/10
1/10
2/10
3/10
AW
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Bayesian Nash Equilibrium

• This yields the following payoff matrix and a
  single pure strategy Nash equilibrium

Example calculation Bayesian Nash Equilibrium
payoff (.4)(-1) (.1)(0) (.2)(28)
(.3)(12) 8.8
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Interpretation of Bayesian Nash Equilibrium

• If Player A is Strong, he takes action a2 and a1
  if Weak.
• Player B takes action b1 irrespective of state.
• Emerged from the known probabilities of each
  possible match-up
• Nash optimal Best response (in a Bayesian
  sense) on the part of each player to the actions
  available to his opponent
• Note that each player has a pure state-dependent
  strategy
• (However, an outside observer could interpret it
  as a mixed strategy, with Nature playing the part
  of a third indifferent player who randomly
  chooses states for players A and B according to
  fixed probability distributions)

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Static Bayesian Game 2 Cournot Model

• Consider a Cournot model comprising two firms A
  and B producing the same commodity to satisfy
  market demand, D.
• The commodity price on the market is given by
• Firm As cost of producing the commodity is cAqA
• cAis the marginal cost
• qA is the quantity that Firm A produces.
• Firm Bs cost of producing the commodity is
• cB1qB, with probability p
• cB2qB with probability (1-p).
• Player state defined by its marginal cost
• Each firm seeks to maximize its profit by
  anticipating the market price

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Cournot Model and Asymmetric Information

• Firm B knows its state and Firm As state
• Firm A knows its own marginal cost but can only
  estimate Firm Bs state
• Each firm knows of the others degree of
  knowledge
• Gibbons calls this a Bayesian game with
  asymmetric information
• Firm A chooses the optimal quantity qA to
  produce
• Firm B chooses the optimal quantity qB to produce
  
• For cB1
• For cB2

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Analytical Solution Bayesian Nash Equilibrium

• System of Equations
• Solutions

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Bayesian Game with Continuous Type Space Auction

• Consider an auction comprising two bidders and
  one item
• Players offer bids, b1 or b2, for the item
• b1 b2 ?0, 1
• Each bidder values the item at v1 or v2 with
  payoff v1 p or v2 p, respectyively
• v1 v2 ?0, 1

Note The latter term in this utility function
applies only when bids are offered in fixed
increments. For bids from the continuous set
0,1, this term is zero.
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Linear Equilibrium

• We simplify the search for equilibrium by
  limiting the solution to the linear form
• bi(vi) ai civi
• This does not limit the player action spaces to
  linear strategies, but simply looks for a linear
  equilibrium solution
• We can assume that a player i will neither bid
  above the expected highest bid nor below the
  lowest expected bid of player j
• Therefore, aj ? bi ? ajcj, since vj?0,1 and is
  a uniformly distributed random variable

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Linear Equilibrium

• This gives us

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Linear Equilibrium

• Since we are looking for a linear solution, ai
  and aj ? 0, since values greater than zero would
  yield a non-linear solution or, if greater than
  1, would yield an infeasible solution since
  neither bidder will offer more than he values the
  item.
• Thus, since the bids must be non-negative, ai
  aj 0, and the solution is that each bidder will
  offer one half his valuation of the item.

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Dynamic Games with Bayesian Players

• Dynamic games with incomplete or imperfect
  information
• Players move after observing the actions taken by
  their opponents.
• Recall from the initial discussion on static
  games that information incompleteness implied an
  information deficit with respect to an opponents
  type or state
• Information imperfection implies that each
  successive players move is based on complete
  information about the state of the other players
  but flawed information about the state of the
  game i.e., the play history on the part of his
  opponents
• These games require a new solution concept
  perfect Bayesian equilibrium

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Perfect Bayesian Equilibrium

• Gibbons gives the following four requirements for
  a perfect Bayesian equilibrium
• For each game turn, the moving player must have a
  belief about the state of the game, i.e. the play
  history to that point, in the form of a
  probability distribution over the set of the
  possible game sub-states at that point.
• Given their beliefs, the players strategies must
  be sequentially rational.
• Note An example of irrational (but effective
  under some circumstances) strategy is tit-for-tat
  in repeated prisoners dilemma games. Axlerod,
  Sigmund
• At each game state on the equilibrium path,
  beliefs are formed by observation-driven Bayes
  rule and players equilibrium strategies.
• (For a given equilibrium in a sequential game, a
  game state is on the equilibrium path if it will
  be reached with positive probability when the
  game is played according to equilibrium
  strategies. Otherwise, the state is off the
  equilibrium path.)
• For game states off the equilibrium path, beliefs
  are formed by Bayes rule and players
  equilibrium strategies where possible.

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Simple Example

• Consider the following 3-player Game Tree. Each
  set of nodes corresponding to outcomes associated
  with any particular players move represents a
  possible game state.

R1. This requirement is relevant for P3 only
since if P1 chooses A, the game is over, and thus
P2 has only to believe that he is in state D if
he has a turn. Player 3 must conclude that p 1
since R is dominated by L for player 2. R2. Given
this belief, Player 3 must choose R. R3. This
requirement is satisfied by R1. R4. This
requirement is trivially satisfied since there
are no states off the equilibrium path. Thus,
the equilibrium (D,L,R) can be confirmed by
inspection.
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Signaling Games

• Games of two players with incomplete information
  about the opponents type
• One player is the Sender, one is the Receiver.
• Nature draws a type for the Sender according to a
  probability distribution on the set of feasible
  types.
• The Sender observes his type and sends a message
  based on that type. The sender can follow
  pooling, separating or hybrid strategies.
• A pooling Sender transmits the same message
  regardless of type.
• A separating Sender always transmits different
  messages for each type.
• The Receiver observes the message but not the
  type and chooses an action.
• Payoffs to the Sender and receiver are each a
  function of Sender type, message and Receiver
  action.

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Requirements for Perfect Bayesian Equilibrium in
Signaling Games

• 1. After observing the Senders message, the
  Receiver must have a belief about the Senders
  type in the form of a probability distribution
  conditional upon the message transmitted.
• 2R. For each message observed, the Receivers
  action must maximize the Receivers expected
  payoff, given the belief about the Senders type.
• 2S. For each type determined by Nature, the
  Senders message must maximize his expected
  payoff, given the Receivers strategy, defined as
  the set of actions to be taken as functions of
  the message transmitted.
• 3. For each message transmittable by the Sender,
  if there exists a sender type such that the
  message is optimal for that type, then the
  Receivers belief about the Senders type must be
  derivable from Bayes rule and the Senders
  strategy.

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Example Job Market Signaling

• Nature determines a workers (the Sender)
  productive ability, which can be either High or
  Low. The probability that his ability is High is
  q.
• The worker observes his ability and chooses a
  level of education (his message to potential
  employers).
• The hiring market (the Receiver) observes the
  workers level of education and, based on a
  belief about the workers ability, offers a wage
  (Receivers action).
• Payoff to the worker is W C(a, e), where W is
  the wage offered, C is the cost (financial
  intellectual difficulty) of attaining a
  particular level of education as a function of
  ability a and education level e. Presumably, the
  cost of attaining a higher level of education for
  a Low ability worker is relatively high due to
  the additional intellectual difficulty sustained
  by the worker in its pursuit.
• Payoff to the hiring market is P(a, e) W, where
  P is the level of productivity supplied by the
  worker as a function of ability and education
  level.

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Complete Information Solution
P(H,e)
IH
IL
W
Note the marginal cost of education is higher for
a Low ability worker, thus he would require a
higher relative salary to justify pursuing a
higher education, hence the steeper indifference
curve. The Productivity lines are found from the
Nash solution W(e) P(?,e) in which the market,
which is presumed to be competitive and therefore
devoid of excess profit, offers a wage equal to
the expected level of productivity.
W(H)
P(L,e)
W(L)
e
e(L)
e(H)
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Pooling Equilibria and the Power of Envy

• Suppose now that the hiring market has incomplete
  information about the workers type and only
  observes the level of education attained by the
  workers.
• Suppose further that a Low ability worker is
  envious of a High ability workers salary and
  decides to attempt to masquerade as a High
  ability worker by getting a more advanced degree.
• This constitutes a pooling strategy since the
  worker will attempt to signal to the hiring
  market that he is of High ability irrespective of
  type.
• Note, this is only rational if the following
  inequality holds
• W(H) - CL,e(H) gt W(L) CL,e(L)

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Masquerading Workers with Pooling Strategies
IH
IL
P(H,e)
W
qP(H,e) (1-q) P(L,e)
Wp
Here the Nash equilibrium sets the wage at wp,
where the expected Productivity line intersects
both indifference curves.
P(L,e)
W(L)
e
e(L)
ep