GAME THEORY
1) Strategic Form or Normal Form Game
2) Extensive Form Game
Normal (Strategic) Form Game
Provide a ‘reduced’ summary of a game. Normal form
game is defined by
- a set of players
- a set of strategies
Common Knowledge
- a set of payoffs
}
Description is common knowledge: each knows that another
person knows that he knows and so on ……….that he
knows …..(e.g. the sets of players, strategies and payoffs)
Players are fully rational, maximize expected utility given
subjective beliefs and beliefs are modified when new
information arrives (e.g. according to Bayes’ law)
Example: Matching Pennies
Column
Head
Tail
Head
Row
Tail
1 , -1
-1 , 1
-1 , 1
1 , -1
No Nash Equilibrium in the game
The entries in each cell
indicate payoffs.
The first number is
Row’s payoff, the second
is Column’s payoff.
Example: The Prisoner’s Dilemma game
Column
Cooperate
Defect
Cooperate
Row
Defect
0,0
1 , -5
-5 , 1
-2 , -2
Each player has an incentive to defect regardless of what the
other does → Defect is a dominant strategy
Strategy profile (Defect, Defect) is the unique Nash
equilibrium in pure strategy.
However, cooperation gives higher payoffs
Note: the first element in a vector of strategies denote the
first player’s (Row’s) strategy, the second element denotes
the second player’s (Column’s) strategy.
Solution Concepts
Pure strategies: Strategies in which a choice is made with
probability of 1
Mixed strategies: Random strategies i.e. choices are made
with probability > 0 and < 1
If R is the set of strategies available to Row,
the set of mixed strategies is the set of all probability
distributions over R where the probability of playing
strategy r in R is pr (similarly for Column)
If Row’s subjective probability distribution over Column’s
set of strategies is ΠC, πc is the prior probability that
Column’s choice is c, and
Column’s subjective probability over Row’s set of strategies
is ΠR, πr is the prior probability that Row’s choice is r
Note that ∑πc = 1 and ∑πr =1
c∈C
r∈R
- Row’s objective is to choose pr for each r to maximize
Row’s expected payoff
∑ ∑ pr πcur (r,c)
r∈R c∈C
- Column’s objective is to choose pc for each c to
maximize Column’s expected payoff ∑ ∑ pcπr uc (r, c)
c∈C r∈R
Nash Equilibrium
Consistency Requirement: each player’s belief about
the other player’s choices coincides with the actual choices
the other player intends to make.
Rational Expectations: Expectations coincide with
actual frequencies.
Definition: Nash Equilibrium
A Nash equilibrium consists of vector of prior belief
probabilities (πr, πc) over strategies and vector of
probabilities (pr, pc) of choosing strategies such that
1. The beliefs are correct: pr = πr and pc = πc
for all r and c and
2. Each player is choosing probability pr or pc (Row
player is choosing pr for all r and Column player is choosing
pc for all c) so as to maximize his own expected utility
given his belief.
Alternatively, with two strategies for each player, a
Nash equilibrium is a pair ( p̂r , p̂c ) such that
Eu r ( pˆ r , pˆ c ) ≥ Eu r ( pr , pˆ c ) for every p r ∈[0,1]
Euc ( pˆ c , pˆ r ) ≥ Euc ( pc , pˆ r ) for every pc ∈[0,1]
*Here the distinction between beliefs and actual frequencies
is blurred.
Definition: Nash equilibrium in pure strategies
A Nash equilibrium in pure strategies is a pair
* *
(r , c ) such that ur(r*, c*) ≥ ur(r, c*) for all Row’s strategies
r and uc(r*, c*) ≥ ur(r*, c) for all Column’s strategies c
In a Nash equilibrium, no player has incentive to
deviate unilaterally from a Nash equilibrium strategy.
→ A ‘Rest’ Point
Example: Battle of the sexes
Ruth Top (Opera)
Bottom (Football)
Chris
Left (Opera) Right (Football)
2, 1
0, 0
0, 0
1, 2
Two Nash equilibria (Top, Left) and (Bottom, Right) in
pure strategies.
Finding the solution by maximization
Row’s problem is to maximize expected payoff
Max
pt, pb
pt [2pl + 0pr] + pb [0pl + 1pr]
such that pt + pb =1
pt ≥ 0 and pb ≥ 0
Lagrangian
L = 2ptpl + pbpr − λ (pt + pb − 1) − µ1 pt − µ2 pb
Differentiate wrt. pt and pb yield
∂L = 2pl − λ − µ1 = 0
∂pt
∂L = 2pr − λ − µ2 = 0
∂pt
Since we already know the Nash strategy profiles in pure
strategies. We will consider only the case where pt > 0
and pb > 0 (i.e. µ1 = 0 and µ2 = 0)
→ pl = 1/3 and pr = 2/3
Similarly, can solve Column’s maximization problem
→ pt = 2/3 and pb = 1/3
Diagram: Best response functions and equilibria
Interpretation of Mixed Strategies
- Some games have strategies that are more sensible as
mixed strategies than others
- For any mixed strategy equilibrium, if one player
believes the other will play the equilibrium mixed
strategies, then he is indifferent as to whether he plays
any other mixed or pure strategies as long as the set of
strategies that form a basis for that strategy belongs to
that of his equilibrium mixed strategy.
- Can interpret mixed strategies as being random chance
of meeting a person from a population who plays a pure
strategy but that strategy is not known.
- Alternatively, individuals’ choices may be deterministic
e.g. depend on moods but there is still uncertainty for
other players.
Repeated Game
- A repeated game is a repetition of a one-shot game
- The strategy space of the repeated game is much larger
than that of a one-shot game
Example: Prisoners’ Dilemma
In the long run, it is in the best interest of both players
to get (Cooperate, Cooperate) as a solution.
Finitely Repeated Game
Last round: They are essentially playing a one-shot
game. Hence, (Defect, Defect)
Next to last: Still no long-run benefit to encourage
cooperation. Again, (Defect, Defect)
Same logic applies to other previous rounds
*Hence, in all periods, (Defect, Defect) is the solution.
Infinitely Repeated Game
Consider a trigger strategy (or punishment strategy)
which specifies Cooperate on the current move unless the
other player defected on any previous rounds
Assume that each player discount their future payoff by
1+ρ, ρ is the discount factor
- Suppose, they cooperate up to move at period T.
At T, if a player decides to defect, he gets 1 and a
stream of payoff -2 forever
1
Expected payoff from deviation = 1 − 2 ( ρ )
- Suppose, they cooperate up to move at period T.
At T, if a player decides to defect, he gets 1 and a
stream of payoff -2 forever
Expected payoff from cooperation = 0
1
Continue to choose Cooperate if 0 > 1 − 2 ( ρ )
i.e. ρ > ½
Refinements of Nash Equilibrium
- Nash (1950) shows that with finite number of players
and finite number of pure strategies, a Nash equilibrium
exists (at least in mixed strategy)1
- But we may have multiple Nash equilibria.
- Refinements of Nash equilibrium are criteria used to
choose among Nash equilibria.
Dominant Strategies
Let r1 and r2 be two of Row’s strategies.
r1 strictly (weakly) dominates r2 if payoff of r1 is strictly
larger for all choices Column might make (at least as large
for all choices and strictly larger for some choices)
A Dominant Strategy Equilibrium is a choice of
strategy by each player such that each strategy dominates
every other strategies.
Elimination of Dominated Strategies
Given a game, one first eliminates all strategies that are
dominated and then calculate the Nash equilibria.
Row Top
Bottom
h
1
Left
2, 2
2, 0
Column
Right
0, 2
1, 1
Also, with infinite continuum of strategies, a Nash can be shown to exist in pure strategy.
Two Nash equilibria (Top, Left) and (Bottom, Right) in
pure strategies.
If Row assumes that Column will never play his weakly
dominated strategy then the only equilibrium for the game is
(Bottom, Right).
Although, elimination of strictly dominated strategies is
generally acceptable, elimination of weakly dominated
strategies may not be sensible.
Sequential Game
Some choices are made sequentially and players may
know other players’ choice before making his own.
Column
Left
Right
1 , 10
1, 5
Row Top
0, 0
2, 1
Bottom
h
Two pure Nash: (Top, Left) and (Bottom, Right)
Extensive Form Game
Provides ‘extended’ description of the game. Often
represented by a game tree indicating the choices that each
player can make at each stage of the game.
1, 10
Left
Column
Top
1, 5 Row moves first,
Right
Column moves after
Row
observing Row’s move.
Left
0, 0
Bottom
Column
Right
2, 1
Each decision point is a node
A subgame consists of a node and all strategies and
payoffs available from then on.
A Subgame Perfect Equilibrium (SPE) is Nash
equilibrium in every subgame. We can usually solve for
SPE using ‘backward induction’ Note that (Top, Left) is
eliminated.
Extensive form can be used to model situations where
some moves are sequential, some are simultaneous.
Information Set is the set of all nodes that cannot be
differentiated by the agent.
1, 10
Left
Column
Top
Right
1, 5
Left
0, 0
Right
2, 1
Row
Bottom
Column cannot
tell where he is,
as if we are in
simultaneous
move game
Column
Example: A simple bargaining model
- N- stage game with 2 players A and B who have $1 to divide
between them
- 1st stage, A makes an offer, B can accept or wait
- 2nd stage, B makes an offer, A can accept or wait
- And so on to Nth stage
Solving for SPE
Suppose that A’s and B’s discount rates are α and β
If N=3, at last stage A offers B $0 and A gets $1. Hence, at 2nd
stage B can offer A $α and gets $(1- α). At 1st stage, A offers B
$β(1- α) and takes $(1- β(1- α)) for himself.
Game with Incomplete Information
Players may not have complete information about other
players, e.g. valuation of good, preferences, etc.
Agent’s type describes all uncertainty that one agent
may have about another.
A Bayes’ Nash equilibrium is a set of strategies for
each type of player such that the expected value of each type
of player is maximized given the strategies of other players.
Example: Battle of the sexes (modified)
- Ruth knows Chris’ preference but Chris does not know
hers.
- Ruth may like to be with him (type 1)
or to be alone (type 2).
- Chris attaches a probability ρ to Ruth being a type 1
- Ruth knows Chris’s estimate of her preference (know ρ)
This is assumption of ‘common knowledge’
- If Ruth is type 1, her preference is as before
- If she is type 2, she gets 0 if she goes anywhere with
Chris and gets 1 if goes alone to football and 2 if goes
alone to opera.
- Chris’ preference is as before
Solving for pure strategy Bayes-Nash equilibrium
First, turn incomplete information game to imperfect
information game
(Draw tree)
The set of pure strategies for Chris is {O, F}
For Ruth is {(O,O), (O,F), (F,O), (F,F)} (1st entry for type 1)
1) Suppose Chris plays F, in a best response, Ruth plays F
if she is type 1 and O if she is type 2. i.e. (F,O)
Chris’ expected payoff is 2ρ + 0(1- ρ) = 2ρ
For this profile ((F,O), F) to be a Bayes’ Nash
equilibrium, Chris must actually prefer F to O i.e.
(expected payoff from F) 2ρ ≥ 1- ρ (expected payoff from O)
→ ρ ≥ 1/3
2) If Chris plays O, Ruth will play (O,F).
For ((O,F), O) to be a Bayes’ Nash,
Chris must actually prefer O to F, that is we must have
1ρ + 0(1- ρ) ≥ 0ρ + 2(1- ρ)
→ ρ ≥ 2/3
Two pure-strategy Bayes-Nash equilibria when ρ ≥ 2/3.
i.e. ((F,O), F) and ((O,F), O)
One pure-strategy Bayes-Nash when 1/3 ≤ ρ < 2/3,
i.e. ((F,O), F)
No pure-strategy Bayes-Nash when ρ < 1/3
Discussion
Different beliefs about distribution of various types lead
to different optimal behabiour.
Ladyard (1986): nearly any pattern of behaviour can be
supported by some pattern of beliefs.