1. 6-1 Bayesian Games: Taste
  2. 6-2 Bayesian Games: First Definition
  3. 6-2 Bayesian Games: First Defintion (yoav)
  4. 6-3 Bayesian Games: Second Definition
  5. 6-4 Analyzing Bayesian Games
  6. 6-5 Analyzing Bayesian Games: Another Example

Game Theory

Week 6: Bayesian Games

https://www.coursera.org/learn/game-theory-1/home/week/6

6-1 Bayesian Games: Taste

例子：拍卖

6-2 Bayesian Games: First Definition

  • Bayesian Game: Information Sets

    贝叶斯博弈

    A Bayesian game is a tuple \((N, G, P, I)\) where

    • \(N\) is a set of agents

    • \(G\) is a set of games with \(N\) agents each such that if \(g, g' \in G\) then for each agent \(i \in N\) the strategy space in \(g\) is identical to the strategy space in \(g'\)

    • \(P \in \prod (G)\) is a common prior over games, where \(\prod(G)\) is the set of all probability distributions over \(G\)

    • \(I = (I_1, \cdots, I_N)\) is a set of partitions of \(G\), one for each agent

6-2 Bayesian Games: First Defintion (yoav)

6-3 Bayesian Games: Second Definition

  • Directly represent uncertainty over utility function using the notion of epistemic type.

  • Bayesian Game: Epistemic Types

    A Bayesian game is a tuple \((N, A, \Theta, p, u)\) where

    • \(N\) is a set of agents

    • \(A = (A_1, \cdots, A_n)\), where \(A_i\) is the set of actions available to player \(i\)

    • \(\Theta = (\Theta_1, \cdots, \Theta_n)\), where \(\Theta_i\) is the type space of player \(i\)

    • \(p : \Theta \mapsto [0, 1]\) is the common prior over types

    • \(u = (u_1, \cdots, u_n)\), where \(u_i : A \times \Theta \mapsto \mathbb{R}\) is the utility function for player \(i\)

6-4 Analyzing Bayesian Games

  • Expected Utility

    期望效用

    3 standard notions:

    1. ex-ante

      the agent know nothing about anyone's actual type

      事前：该代理不知道其它代理的实际类型

    2. interim

      an agent knows her own type but not the types of the other agents

      中期：该代理知道自己的类型，但不知道其它代理的类型

    3. ex-post

      the agent knows all agents' types

      事后：该代理知道所有代理的类型

Given a Bayesian game \((N, A, \Theta, p, u)\), where \(i\)’s type is \(\theta_i\) and where the agents’ strategies are given by the mixed strategy profile \(s\)

  • Interim expected utility

    \[EU_i(s \mid \theta_i) = \sum_{\theta_{-i} \in \Theta_{-i}} p(\theta_{-i} \mid \theta_i) \sum_{a \in A} \left( \prod_{j \in N} s_j(a_j \mid \theta_j) \right) u_i(a, \theta_{-i}, \theta_i)\]

  • Ex-ante expected utility

    \[EU_i(s) = \sum_{\theta_{-i} \in \Theta_{-i}} p(\theta_{-i}) EU_i(s \mid \theta_i)\]

  • Ex-post expected utility

    \[EU_i(s, \theta) = \sum_{a \in A} \left( \prod_{j \in N} s_j(a_j \mid \theta_j) \right) u_i(a, \theta)\]

  • Best response

    \[BR_i(s_{-i}) = \text{argmax}_{s'_i} EU_i(s'_i, s_{-i} \mid \theta_i)\]

    if \(\forall \theta_i \in \Theta_i, p(\theta_i) \gt 0\)

    \[BR_i(s_{-i}) = \text{argmax}_{s'_i} EU_i(s'_i, s_{-i}) = \text{argmax}_{s'_i} \sum_{\theta_i} EU_i(s'_i, s_{-i} \mid \theta_i)\]

  • Bayesian equilibrium / Bayes-Nash equilibrium

    \[\forall i, s_i \in BR_i(s_{-i})\]

6-5 Analyzing Bayesian Games: Another Example

例子：A Sheriff's Dilemma