Bayesian Games

Motivation

  • auctions
    • I'm not quite sure what are the utilities of other players
  • usually everyone knows
    • the number of players
    • the actions available to each player
    • payoff associated with each action vector

Assumptions

  • there are some games (not only one)
    • all games have the same number of agents
    • the same strategy space for each agent
    • the only difference is in the payoff
  • agents' beliefs are posterior
    • obtained by conditioning a common prior on individual private signals
    • common prior - what's possible
    • individual private signals - beliefs that agents have

Bayesian game

  • Definition 1: based on Information set
    • Intuition
      • BG - a set of games that differ only in their payoffs
      • plus a common prior is defined over them
      • and a partition structure is defined over the games for each agent
    • A BG is a tuple [math](N, G, P, I)[/math] where
      • [math]N[/math] - a set of agents
      • G[math][/math] set of games with N agents for each
        • if [math]g[/math] and [math]g' \in G[/math] for each agent [math]i \in N[/math]
        • the strategy space in [math]g[/math] is identical to the strategy space in [math]g'[/math]
        • (so games differ only in their utility functions)
      • [math]P \in \Pi (G)[/math] - a common prior over games
        • [math]\Pi(G)[/math] set of probability distribution over [math]G[/math]
        • (how likely each of these games is)
      • [math]I = (I_1, I_2, ..., I_N)[/math] is a set of partitions of [math]G[/math], one for each agent
        • set of equivalence classes: some games are indistinguishable
    • Example
      • 4 games
        • Matching Pennies
        • Prisoner's Dilemma
        • Coordination game
        • The Battle of the Sexes
      • equivalence classes
        • Player I
          • MP and PD
          • Coord and BoS
        • Player II
          • MP and Coord
          • PD and BoS
      • when playing, players don't know what game they're playing
      • only the equivalence class
  • Definition 2: based on epistemic types
    • directly represents uncertainly over utility function using the notion of epistemic type
    • epistemic type - private information of an agent
    • A BG is a tuple [math](N, A, \Theta, p, u)[/math] where
      • [math]N[/math] - a set of agents
      • [math]A = (A_1, A_2, ..., A_n)[/math]
        • [math]A_i[/math] - set of actions available to [math]i[/math]
      • [math]\Theta = (\theta_1, ..., \theta_n)[/math]
        • [math]\theta_i[/math] - type space of player [math]i[/math]
      • [math]p: \theta \mapsto [0, 1][/math]
        • the common prior over types
      • [math]u = (u_1, ..., u_n)[/math]
        • where [math]u_i = A * \theta \mapsto \mathbb{R}[/math]

Analysing Bayesian Games

  • Bayesian (Nash) Equilibrium
    • a plan of actions for each player as a function that maximizes each type's expected utility
    • so it should be a best reply
    • If I observe a certain type, what am I going to do?
    • expecting over the actions of other players
      • what are the expected action distributions we're going to face
    • expecting over the types of ther players
  • Strategies
    • given a Bayesian finite game [math](N, A, \Theta, p, u)[/math]
    • pure strategy
      • [math]s_i : \theta_i \mapsto A_i[/math]
        • for a type, what action you'll take?
      • a choice of a pure strategy for player i as a function of her type
    • mixed strategy
      • [math]s_i : \theta_i \mapsto \Pi(A_i)[/math]
        • [math]\Pi(A_i)[/math] - probability distribution over actions of your type
      • a choice of x mixed action for player i as a function of his type
      • distribution over actions
        • [math]s_i(a_i | \Theta_i)[/math]
        • [what's the probability that action a_i will be chosen if they happen to be of type [math]\Theta_i[/math]]
        • the probability under mixed strategy [math]s_i[/math] that agent [math]i[/math] plays action [math]a_i[/math], given that type is [math]\Theta_i[/math]
  • types
    • ex-ante
      • the agent knows nothing about anyone's actual type
    • interim
      • agents know their own types, but don't know the types of each other
      • for player i with respect to type \theta_i and mixed strategy profile s
      • expected utility
        • [math]EU_i(s | \Theta_i) = \sum_{\theta_{-i} \in \Theta_{-i}} p (\theta_{-i} | \theta{i}) * \sum_{a \in A}(\prod_{j \in N} s_i(a_i | \theta_i) * u_i(a, \Theta_i, \Theta_{-i}))[/math]
        • [math]u_i(a, \Theta_i, \Theta_{-i})[/math] - utilities evaluated with respect to their types
        • [math]\prod_{j \in N} s_i(a_i | \theta_i)[/math] - what other players will be doing
        • [math]\sum_{\theta_{-i) \in \Theta_{-i}} p (\theta_{-i} | \theta{i})[/math] - sum across all probabilities of types for others
        • [math]EU_i(s | \Theta_i)[/math] - what can i expect of he of type \Theta_i and follows s
    • ex-post
      • averybody knows everything
      • expected utility
        • [math]EU_i (s) = \sum_{\theta_i \in \Theta_i} p(\theta_i) EU_i(s | \theta_i)[/math]
  • Bayesian Equilibrium
    • a mixed strategy profile s that satisfies
    • [math]s_i \in \arg \max_{s'_i} EU_i(s'_i, s_{-i} | \Theta_i)[/math]
    • each individual should choose the best response, maximizing the expected utility
    • for each [math]i[/math] and [math]\theta_i \in \Theta_i[/math]
    • summary
      • it explicitly models behavior in uncertain environment
      • players choose strategies to maximize their payoffs in response to others
      • accounting for
        • strategic uncertainty about how others will play
        • payoff uncertainty about the value of their actions
  • A Sherif's Dilemma
    • a sheriff faces an armed suspect and they each must (simultaneously) decide whether to shot or not
    • a suspect is criminal with probability p and not a criminal with probability 1 - p
    • the sheriff would rather shoot if suspect shoots, and not shoot otherwise
    • the criminal would rather shoot even if the sheriff doesn't - he doesn't want to be caught
    • the innocent would rather not shoot even if the sheriff does
    • ==> sheriff's best reply to shoot if [math]p \gt 1/3[/math]


Sources