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