Games of Incomplete Information These games drop the assumption that players know each other’s preferences.

Complete info: players know each other’s preferences Incomplete info: some players do not know some of the other’s preferences (types)

Although the analysis is more complicated, we still impose fair amount of structure:1) Players have types 2) Probability distributions associated with types 3) An extra player, Nature, is included to resolve uncertainty

about types

Each player is typically assumed to observe his or her own preferences (or type)

Assume a player’s preferences are determined by the realization of some random variable

Other players don’t observe the realization of the RV but the ex ante probability distribution of the RV is assumed to be common knowledge

Probability distribution of types

Nature moves first in the extensive form game and determines players types

type

Impose structure through assumptions on the distribution of types: , ,

is distributed according to some probability distribution, which all players know.

Since players know

i i i i

i

U s s

the probability distribution, we can use expected utilities i

The simplest games assumes two possible types: , with probabilities , 1-H L p p

Consider a modified version of chicken: -player (1)’s payoffs are common knowledge (maybe he’s a famous chicken contestant)-player (2)’s payoffs are known only to (2)

Assume (2) can be one of two types – mean or not

If (2) is mean, - he hates to lose even more than he hates death

If (2) is not mean, - he does not care about winning or not - he just does not want to die. - he is happiest when both turn .

Chicken w/ Incomplete Information

21

turn don’t turn

turn 2, 3 1, 2

don’t turn

4, 2 0, 0

21

turn don’t turn

turn 2, 0 1, 4

don’t turn

4, -1

0, 0

Game tree with probabilities Now consider the probability distribution over types

Suppose (2) is mean with probability μ and not mean with probability (1- μ)

(1) doesn’t know whether

(2) is mean or wimpy(2) knows he is wimpy but doesn’t know what (1) has done

(2) knows he is mean, but doesn’t know what (1) has done

More contingencies

A strategy for (2) is now a contingent plan that takes natures move into account.

(2)’s possible strategies are: turn if mean, turn if not turn if mean, not if not not if mean, turn if notnot if mean, not if not

(1)’s strategy set is just like in the case of complete information: turn

this is the only choice not

Incomplete information adds contingencies to strategies - remember, that strategies describe what to do on and off the path

Bayesian Game

Games of Incomplete Information are often referred to as Bayesian Games

Formally, each player in a Bayesian game has a payoff function

, , where is a RV choose by nature and observedi i i i iU s s

1 2

1 2

only by .

We require that where is a set known to all players

Assume the joint probability distribution of , ,... ,

one for each player, is known to all: , ,... ,

i i i

n

n

i

F

1 2 1 2

where is the probability distribution over possible vectors of player's types

Let = ... be the set of all possible combinations of , ,... . is the set of all possible vec

i n n

F

tors of player types

A Bayesian Game is summarized by , , , ,B i iN s U F

Bayesian Nash Equilibrium

Defn: A pure strategy for in a Bayesian game is a decision rule ,

that specifies player ' choice for each possible type he might be.

Defn: Player ' strategy set is the set of all po

i ii s

i s

i s

1 1 1 1

tak

ssible functions . Denote this set by .

Defn: Player ' expected payoff given a profile of pure strategies for all players

,... is given by ,... ,... ;

i i i

n i n i n n i

s

i s

s s U s s E U s s

e the expectation with respect to

1 1

Defn: A pure strategy Bayesian Nash equilbrium for the Bayesian game

, , , , is a profile of decision rules ,...

that constitutes a

i i n nN s U F s s

Nash equilibrium of game , , .

That is, for every 1,... , , , for all

N i i

i i i i i i i i

N U

i N U s s U s s s

How to compute a Bayesian NE

1 1Prop: A profile of decision rules ,... is a Bayesian Nash equilbrium in

Bayesian game , , , , if and only if for all and all

occuring with positive probability

i

n n

B i i i i

i

s s

N s U F i

E U s

, ; | , ; |

for all , where the expectation is taken over realizations of the other players'

types conditional on player 's realization of his type .

,

ii i i i i i i i i i i

i i

i

i i i i

s E U s s

s S

i

U s s

; says preferences depend upon types

indicates a) player know own type

b) player ' knowledge of own type could provide info on other's typesi

i i

i iE U i

i s

Solving for the Bayesian NE In the Incomplete Information Chicken Game

If (2) is mean (2) has a strictly dominant strategy – don’t turn If (2) is not (2) has a strictly dominant strategy – to turn

So (2)’s optimal strategy is – not if mean – turn if not mean

Player (1)s best response depends on μ (μ is the probability that (2) is mean)

1

1

(1)s expected payoff from turning, given (2)s strategy is 1 2 1

(1)s expected payoff from not turning is 0 4 1

2Turning is a BR if 1 +2 1- 4 1- , 3

2If BR is turn ; if3

1 12 2 BR is not ; if BR is indifferent3 3