Information in Games Games with complete information

each player knows the strategy set of all opponents

each player knows the payoff of every opponent for every possible (joint) outcome of the game

Strong assumption

Information in GamesGames with complete information

each player knows the strategy set of all opponents

each player knows the payoff of every opponent for every possible (joint) outcome of the game

Strong assumptionDoes not happen often in real world

e.g. 1: competing sellers don’t know payoffs for each other

e.g. 2: corporation bargaining with union members don’t know the (dis)utility of a month’s strike for the union

Incomplete InformationGames in real world scenarios have

incomplete informationHow do we model incomplete information?

Incomplete InformationGames in real world scenarios have

incomplete informationHow do we model incomplete information?Consider a player’s beliefs about other

players’ preferences, his beliefs about their beliefs about his preferences, …

Modeling Games with Incomplete InformationHarsanyi’s approach:

Each player’s preferences are determined by a random variable

The exact value of the random variable is observed by the player alone

But…

Modeling Games with Incomplete InformationHarsanyi’s approach:

Each player’s preferences are determined by a random variable

The exact value of the random variable is observed by the player alone

But…the prior probability distribution of the random variable is common knowledge among all players

Modeling Games with Incomplete InformationIncomplete information now becomes

imperfect informationWho determines the value of the random

variable for each player?

Modeling Games with Incomplete InformationIncomplete information now becomes

imperfect informationWho determines the value of the random

variable for each player?Nature

background/culture/external effects on each player

Example: Modified DA’s BrotherPrisoner 2 now can behave in one of two

wayswith probability he behaves as he did in the

original gamewith probability 1- he gets “emotional”, i.e.,

he gets an additional payoff of –6 if he confesses (rats on his accomplice)

Who determines ?“nature” of player 2

Note: Player 1 still behaves as he did in the original game

Type I (prob ) Type II (prob 1-)

Example: Modified DA’s BrotherPlayer 2 has 4 pure strategies now

confess if type I, confess if type IIconfess if type I, don’t confess if type IIdon’t confess if type I, confess if type IIdon’t confess if type I, don’t confess if type II

Player’s strategy is now a function of its type

Note: Player 2’s type is not visible to player 1, Player 1 still has two pure strategies as before

Bayesian Nash GamePlayer i’s utility is written as:

ui(si,s-i, i), where i Є i

i: random variable chosen by nature i: distribution from which the random

variable is chosenF(1,2,3,… I): joint probability distribution

of all playerscommon knowledge among all players

1 X 2 X 3 …X I

Bayesian Nash Game = [I, {Si}, {ui(.)}, F(.)]

Example: Bayesian Nash GameModified DA’s Brother example

1={1} (some constant element because P1 does not have types}

2 ={emotional, not emotional}= 1 X 2 = {(emotional), (non-emotional)}F((some probability values that

give the probabilities of {(emotional), (non-emotional)} This is common knowledge

Bayesian Nash GamePure strategy of a player is now a function of

the type of the player:si(i): called decision rule

Player i’s pure strategy set is S i which is the set of all possible si(i)-s

Player i’s expected payoff is given by:ûi(s1(.), s2(.), s3(.), ….sI(.))=E[ui(s1(1)...sI(I), i)]

Bayesian Nash Game as Normal Form GameWe can rewrite the Bayesian Nash game

[I, {Si}, {ui(.)}, F(.)]

Bayesian Nash Game as Normal Form GameWe can rewrite the Bayesian Nash game

[I, {Si}, {ui(.)}, F(.)]

as[I, {S i }, ûi(.)}]

Definition: Bayesian Nash EquilibriumA pure strategy Bayesian Nash equilibrium

for the Bayesian Nash game [I, {Si}, {ui(.)}, F(.)] is a profile of decision rules (s1(.), s2(.), s3(.), ….sI(.)) that constitutes a Nash equilibrium of game [I, {S i }, {ûi(.)}]. That is, for every i=1…I

ûi(si(.), s-i(.)) >= ûi(s’i(.), s-i(.))

for all s’i(.) Є S i where ûi(si(.), s-i(.)) is the expected payoff for player i

PropositionA profile of decision rules (s1(.), s2(.), s3(.),

….sI(.)) is a Bayesian Nash equilibrium in a Bayesian Nash game [I, {Si}, {ui(.)}, F(.)] if and only if, for all i and all ’i Є i occurring with positive probability

E-i[ui(si(’i), s-i(-i), ’i) | ’i)] >= E-i[ui(si’, s-i(-i), ’i) | ’i)]

for all si’ Є Si, where the expectation is taken over realizations of the other players’ random variables conditional on player i’s realization of signal θ’i.

Outline of Proof (by contradiction)Necessity: Suppose the inequality (on

last slide) did not hold, i.e.,E-i[ui(si(’i), s-i(-i), ’i) | ’i)] <

E-i[ui(si’, s-i(-i), ’i) | ’i)]→ Some player I for whom ’i Є i happens

with a positive probability, is better off by changing its strategy and using si’ instead of si(’i)→ (s1(.), s2(.), s3(.), ….sI(.)) is not a Bayesian Nash

equilibrium→ Contradiction

Outline of Proof (by contradiction)Reverse direction:E-i[ui(si(’i), s-i(-i), ’i) | ’i)] >=

E-i[ui(si’, s-i(-i), ’i) | ’i)] holds for all ’i Є i occurring with positive probability→ player i cannot improve on the payoff received

by playing strategy si(.)

→ si(.) constitutes a Nash equilibrium

Type I (prob ) Type II (prob 1-)

Solution to Modified DA’s BrotherPrisoner 2, type I plays C with probability 1

(dominant strategy)Prisoner 2, type II plays DC with probability

1(dominant strategy)Expected payoff to P1 with strategy DC: -10

+0(1-)Expected payoff to P1 with strategy C: -5 -1(1-)Player 1:

Play DC when -10 +0(1-) > -5 -1(1-)

or, <1/6Play C when 1/6 1/6 makes player 1 indifferent

Weakly Dominated StrategyProblem of weakly dominated strategy:

for some strategy of the opponent the weakly dominant and the weakly dominated strategy give exactly same payoff

player CAN play weakly dominated strategy for some strategy of the opponent

U

D

(D,R) is a Nash equilibrium involving a play of weakly dominated strategies

Perturbed GameStart with normal form game

N=[I, {Si)}, {ui(.)}]Define a perturbed game

[I, {Si)}, {ui(.)}]

where (Sj)={(i: i > i(si) for all si Є Si and si Є

Si i (si)=1}

i(si) denotes the minimum probability of player I playing strategy si

Perturbed GameStart with normal form game

N=[I, {Si)}, {ui(.)}]Define a perturbed game

[I, {Si)}, {ui(.)}]

where (Sj)={(i: i > i(si) for all si Є Si and si Є Si i (si)=1}

i(si) denotes the minimum probability of player I playing strategy si

i(si) denotes the unavoidable probability of playing strategy si by mistake

Definition: Nash equilibrium in Trembling Hand Perfect GameA Nash equilibrium of a game N=[I, {Si)},

{ui(.)}] is (normal form) trembling hand perfect if there is some sequence of perturbed games {k}k=1

∞that converges to N [in the sense that lim k →∞(i

k(si) )=0 for all I and si Є Si ] for which there is some associated sequence of Nash equilibria {k} k=1

∞ that converges to (i.e. such that lim k →∞ k = )

Proposition 1A Nash equilibrium of a game N=[I, {Si)},

{ui(.)}] is (normal form) trembling hand perfect if and only if there is some sequence of totally mixed strategies {k} k=1

∞ such that lim k →∞ k = and i is a best response to every element of sequence {k

-i} k=1∞ for all

i=1…I.

Proposition 2If = (1, 2, 3,.. I) is a normal form

trembling hand perfect Nash equilibrium then is not a weakly dominated strategy for any i=1…I. Hence, in any normal form trembling hand perfect Nash equilibrium, no weakly dominated pure strategy can be played with positive probability.