Game Theory II
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Mixed Strategies (Recap)
• From last week’s lecture…
A mixed strategy for player 𝑖 is a probability distribution 𝜎𝑖 ∈ Δ𝑆𝑖 .
Where for any pure strategy in the strategy set, 𝑠𝑖 ∈ 𝑆𝑖 , 𝜎𝑖 𝑠𝑖 ∈ 0,1 denotes the
probability of player 𝑖 playing the pure strategy 𝑠𝑖 in the mixed strategy 𝜎𝑖 .
• Pure strategies are special cases of mixed strategies: they are degenerate mixed strategies.
• Strategies which involve actual randomization are non-degenerate mixed strategies.
For simplicity, when we say mixed strategy, we refer to non-degenerate ones.
Mixed Nash Equilibria
Mixed Strategy Nash Equilibrium
• With mixed strategies, we can state similar definitions for a NE.
A strategy profile 𝜎1 … 𝜎𝑛 is a (mixed strategy) Nash equilibrium if for each
individual 𝑖, 𝑢 𝜎𝑖 , 𝜎−𝑖 ≥ 𝑢 𝜎𝑖′, 𝜎−𝑖 for any 𝜎𝑖 ′ ∈ Δ𝑆𝑖 .
• All we have done is replace pure strategies 𝑠𝑖 with mixed strategies 𝜎𝑖 .
• Mixed strategy Nash Equilibria can be solved for in a similar manner:
By getting the best response correspondences!
Note that henceforth, we will be referring to Nash Equilibria as those involving all kinds of
strategies: degenerate (pure) and/or non-degenerate mixed strategies.
Matching Pennies Again
Player 2
Heads
Player 1
Heads
Tails
−1,
1,
Tails
1
−1
1,
−1,
−1
1
• Now, if we allow strategies to include randomization between pure strategies, then
there is a Nash equilibria in mixed strategies in the matching pennies game.
• In the matching pennies game , there is a (non-degenerate) mixed strategy NE where
each chooses Heads and Tails with probability ½.
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Solving for all NE
• Now, we will see how to solve for all NE in two player games with two actions.
• We assume that players maximise expected payoffs/utility.
• Consider the following battle of sexes game:
Wife
Football
Husband
Opera
Football
10,
5
0,
0
Opera
0,
0
5,
10
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Solving for all NE
𝑞
Football
Husband
𝑝
Football
1 − 𝑝 Opera
Wife
1−𝑞
Opera
10,
5
0,
0
0,
0
5,
10
Recall that a mixed strategy is any probability distribution between the pure strategies.
• Let the Husband’s strategy 𝜎ℎ be to choose Football with probability 𝑝 and Opera with
probability 1 − 𝑝; 𝑝 ∈ [0,1]
• Let the Wife’s strategy 𝜎𝑤 be to choose Football with probability 𝑞 and Opera with
probability 1 − 𝑞; 𝑞 ∈ [0,1]
Let us solve for the best responses of each of them!
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Solving for all NE (Husband’s BR)
Suppose Wife has the strategy 𝝈𝒘 : choose Football with probability 𝒒
• Given the Wife’s strategy,
• The expected payoff of choosing Football for Husband is 𝑞 10 + 1 − 𝑞 0
• The expected payoff of choosing Opera is for Husband 𝑞 0 + 1 − 𝑞 5
Choosing Football is a best response for Husband if and only if
𝑞 10 + 1 − 𝑞 0 ≥ 𝑞 0 + 1 − 𝑞 5
𝑞 ≥ 1/3
Choosing Opera is a best response for Husband if and only if
𝑞 10 + 1 − 𝑞 0 ≤ 𝑞 0 + 1 − 𝑞 5
𝑞 ≤ 1/3
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Solving for all NE (Husband’s BR)
• The Husband’s best response correspondence to any mixed strategy of his Wife 𝑞 is:
1
• If 𝑞 > , playing Football is a best response for Husband; i.e. 𝑝 = 1.
3
1
• If 𝑞 < , playing Opera is a best response for Husband; i.e. 𝑝 = 0.
3
1
• If 𝑞 = , playing either Opera or Football is a best response for Husband; i.e. 𝑝 ∈ [0 , 1]
3
• Remember 𝑝 is the probability of choosing Football in the Husband’s mixed strategy.
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Solving for all NE (Wife’s BR)
Likewise, suppose that Husband is playing 𝝈𝒉: choose Football with probability 𝒑
Choosing Football is a best response for Wife if and only if
𝑝 5 + 1 − 𝑝 0 ≥ 𝑝 0 + 1 − 𝑝 10
𝑝 ≥ 2/3
And so on…
• The Wife’s best response correspondence to any strategy of her husband, 𝑝
2
• If p > 3, playing Football is a best response for Wife; i.e. 𝑞 = 1.
2
• If p < 3, playing Opera is a best response for Wife; i.e. 𝑞 = 0.
2
3
• If p = , playing anything is a best response for Wife; i.e. 𝑞 ∈ [0 , 1]
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Solving for all NE
• The best response correspondences can be
plotted out in a graph.
• The intersection points indicate which pair of
strategies (𝑝, 𝑞) are best responses to each other.
• Here we have 3 NE: (Football, Football), (Opera,
Opera) and
2 1
,
3 3
.
• The first two are pure strategy NE, while the last
is a mixed strategy NE.
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Applications
Classes of Games.
• There are two interesting classes of games applied to the study of many topics.
• Games of Coordination
• Payoff to players are highest if they can coordinate the sets of actions which they take.
E.g. Battle of sexes game.
• Games of Competition
• Gains for one player mean losses for another and players will never agree on the sets of actions which
they should take. Usually involve only mixed strategy Nash equilibria.
• Here we discuss several other examples of each.
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Games of Coordination : Chicken game
• An example of a game of coordination is the Chicken game where there are two “Utilitarian”
Nash equilibria (Straight, Swerve) and (Swerve, Straight).
• Notice that there is no Pareto Ranking between the two.
Player 2
Swerve
Player 1
Swerve
Straight
0,
1,
Straight
0
−1,
1
−1
−2,
−2
• Even if they can agree that a utilitarian outcome is better, how will it be chosen?
• Sometimes, information unrelated to payoffs in the game can be the deciding tie breaker!
• Social and cultural norms create expectations which can create coordination on specific equilibria.
• E.g. First come – First served, Gender norms…
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Games of Coordination: Chicken game
• There is also a mixed strategy NE!
• If a specific utilitarian NE is not “focal”, then the mixed NE is a likely outcome.
• Chance of getting the very bad outcome (Straight, Straight).
• This kinds of bad situation may be avoided by taking steps to changing the structure of
the game. E.g. Player 1 can commit to going straight, perhaps by:
1. Locking his steering wheel for the other to see (changing his action set)
2. Purchasing accident insurance which is common knowledge. (changing his payoffs)
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Games of Coordination : Assurance Games
• An assurance game is a simultaneous game with multiple Nash equilibria, one of which pareto
dominates the others. A common example is the arm’s race problem.
Russia
Don’t
US
Stockpile
Don’t
5,
5
1,
4
Stockpile
4,
1
3,
3
• There are 3 Nash Equilibria here: (Don’t, Don’t), (Stockpile, Stockpile) and a mixed strategy
(Stockpile with probability 2/3, Stockpile with probability 2/3).
• The pareto dominant outcome (Don’t, Don’t) is not the unique NE here!
• BUT if Russia can assure US that it will not stockpile, then can get to that equilibria.
E.g. Russia moves first and opens itself to inspection --- more on sequential games later!
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Games of Coordination: Prisoner’s Dilemma
• A prisoner’s dilemma game reflects a scenario where each player has a dominant strategy
which leads to failure to coordinate on the efficient outcome.
Clyde
Silent
Bonnie
Confess
Silent
−5,
−5
−30,
−1
Confess
−1,
−30
−10,
−10
• Confess is a dominant strategy here for each, so the unique NE outcome is (Confess, Confess).
• This outcome is pareto dominated by (Silent, Silent): it is “inefficient”!
• Moving sequentially will not solve this problem…
But, if we expand the game and play it repeatedly, then possible to have some punishment which enforces
cooperation. More on repeated games later!
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Games of Competition: Constant Sum Games
• A constant sum game is a simultaneous play game where an increase in payoff to one
player is associated with an exact decrease in payoff to the other play.
• Related to zero sum games. (Sum of entries in each box is always fixed)
Player 2
Left
Player 1
Up
Down
0,
3,
Right
0
2,
−2
−3
1,
−1
• In two action games, given the conflicting interests, Nash Equilibria are mostly in mixed
strategies. (E.g. when there are no dominant strategies for each player)
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Games of Coexistence
• Modifications to assumptions allow for such games to be analysed in different contexts.
• The following simple modifications will allow us to apply game theory to the study of
evolutionary forces:
1. Behaviour are genetically programmed
• Different types select different strategies
2. Evolution selects the population with higher fitness
• Types with higher payoffs will reproduce faster
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Games of Coexistence : Hawk-Dove game
• A famous game of animal interaction is the Hawk-Dove game
Player 2
Hawk
Player 1
Hawk
−2,
Dove
0,
Dove
−2
4,
0
4
2,
2
• Standard interpretation: Hawk and Dove are different strategies − similar to chicken game.
• Alternative interpretation: Hawk and Dove are different genetic types: aggressive/friendly.
• Suppose there is a population 𝑝 of Hawks and 1 − 𝑝 of Doves.
• The expected payoff of Hawks is −2𝑝 + 4 1 − 𝑝 = 4 − 6𝑝
• The expected payoff of Doves is 2 1 − 𝑝 = 2 − 2𝑝
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Games of Coexistence : Hawk-Dove game
• To model fitness, we assume that the population with a higher expected payoff will rise in
population proportion.
• Under these assumptions, 𝑝 = 0.5 is the only stable equilibrium. This is called an Evolutionary Stable
Strategy. It is also equal to the Mixed Strategy Nash Equilibrium of the standard static game.
Payoff
Payoff of Hawks
Payoff of Doves
Hawk population rises
0.5
Dove population rises
𝑝 (Fraction of Hawks)
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Review Questions 1
Consider finite games where a player has a finite number of (pure) strategies
and there are finite players.
Scan here
to participate!
1.
There is always a Nash Equilibrium in pure strategies. (T/F)
2.
There is always a Nash Equilibrium in (degenerate/non-degenerate) mixed
strategies. (T/F)
3.
A pareto efficient outcome will always be achieved in one of the Nash Equilibria.
(T/F)
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Sequential Games
Sequential Games: who plays when?
• So far, we have only analysed games where players moved simultaneously.
These are called simultaneous games.
• But there are other games where players move one after another.
These are called sequential games.
• In two-player sequential games, we call the player who plays first the 1st mover and the
player who plays second the 2nd mover.
• Notice that the possible actions the 2nd mover can make might actually depend on the what the 1 st
mover does!
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Perfect vs Imperfect Information
• A game is said to have perfect information if players move one at a time and the
player at each move knows exactly what moves have occurred thus far in the game.
• Otherwise, a game is said to be of imperfect information.
• Any static (simultaneous move) game can be interpreted as a game of imperfect information. Why?
• If players move one after another, but do not observe each other’s moves, to the players it is as if
they are playing simultaneously.
• The tools we have learnt can be applied to such games of imperfect information.
For sequential games, we will focus on games of perfect information.
Sequential Games:
• There is a major difference between simultaneous games and sequential games
• Even if the actions that the 2nd mover can choose are the same no matter what the 1 st
mover does, the set of possible strategies is not exactly the same!
• This is because when there is perfect information, the 2nd mover can observe the 1st
mover’s action and can also condition his action on it!
• The second mover can have an independent strategy for each action of the first mover.
• Generally, specified strategies of players must be such that a third party can play on
their behalf no matter what happens.
• This will influence the set of Nash equilibria as we will see.
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Sequential Game of Entry
• Consider an Entrant, who decides whether to enter a market. There is an Incumbent who
decides whether to fight or not to fight.
(Fighting can be taken to be some form of taking competitive action, say by raising output beyond standard levels.)
• The first mover, Entrant makes his choice which is observed before the second mover, the
Incumbent makes his move.
• The payoffs are given by the following payoff matrix.
Incumbent
Fight
Entrant
Enter
Stay out
−5,
0,
Not Fight
−5
−5
2,
2
0,
10
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Extensive Form of the Sequential Game
• The game together with the timings of movement can be represented in a game-tree
form or extensive form.
Entrant
Incumbent
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Static game of Entry
• Consider an Entrant, who decides whether to enter a market. There is an Incumbent who again
decides whether to fight or not to fight.
• Suppose that there is imperfect information: the incumbent cannot observe the Entrant’s decision
before making his decision. (Equivalently, this is a simultaneous game)
• The payoffs are again given by same payoff matrix.
Incumbent
Fight
Entrant
Enter
Stay out
−5,
0,
Not Fight
−5
−5
2,
2
0,
10
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Extensive Form of the Static Game
• The simultaneous version can also be represented in extensive form.
• The incumbent does not know whether the entrant enters or stays out before making his decision.
This is represented by the blue dotted line.
• The line connecting the nodes after Enter and Stay Out tells us they are in the same information set:
the incumbent cannot distinguish between these two nodes.
Entrant
Incumbent
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Normal Form of the Static Game
• The Incumbent and Entrant each have 2 possible pure strategies in the static game.
• The normal form for the static game is just the same as the payoff matrix.
Incumbent
Fight
Entrant
Enter
Stay out
−5,
0,
Not Fight
−5
−5
2,
2
0,
10
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Normal Form of the Sequential Game
• The payoff matrix is not the normal form of the sequential game!
• It masks the fact that the second mover can condition his strategy on the first movers’ action.
• The incumbent’s strategy is actually a pair of actions: (X if Entrant enters, Y if Entrant does not enter)
• The incumbent’s strategy in each case will affect the best response of the entrant!
• The “correct” normal form which incorporates this is:
Incumbent
Entrant
F, F
F, NF
NF,F
NF,NF
Enter
-5,-5
-5,-5
2,2
2,2
Stay Out
0,-5
0,10
0,-5
0,10
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Review Questions 2
Consider a game where Player B moves after observing Player A’s move.
Player A can choose either L or R while Player B can choose either T, M, or B.
Scan here
to participate!
1. How many pure strategies does Player A have in his strategy set?
2. How many pure strategies does Player B have in his strategy set?
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Nash Equilibria in Sequential Games
Sequential Entry Game: Nash Equilibria
Incumbent
Entrant
F, F
F, NF
NF,F
NF,NF
Enter
-5,-5
-5,-5
2,2
2,2
Stay Out
0,-5
0,10
0,-5
0,10
• There are three pure strategy NE: {Enter, (NF,F)} ; {Enter, (NF,NF)}; and {Out; (F,NF)}.
• However, are all of them natural?
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Sequential Games: Example
• In the last NE: {Out; (F,NF)}, the entrant stays out because the incumbent’s strategy
(F,NF) “threatens” to fight if he enters.
• But, if the Entrant had entered, this would hurt the incumbent as well and is an empty threat.
Entrant
Incumbent
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Sequential Games: Example
• The reason why this occurs is because the incumbent’s choice of fighting if the Entrant enters is
off the equilibrium path (since the entrant stays out).
• A strategy which is off the equilibrium path does not actually affect payoffs!
Incumbent
Entrant
F, F
F, NF
NF,F
NF,NF
Enter
-5,-5
-5,-5
2,2
2,2
Stay Out
0,-5
0,10
0,-5
0,10
If Entrant Chooses to enter and Strategy for if Entrant Stays Out
changes, but Incumbent’s payoff does not change.
• We hence need a stronger equilibrium notion to rule out such non-credible threats!
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Summary
• For simple two player static games, the different kinds of payoffs that one puts in can be used to
model many different kinds scenarios.
• Often strategic interactions can lead to possibly inefficiencies in outcomes.
• Relying on norms to coordinate beliefs or changing payoffs/actions inherent in the game might be
one solution to avoid these inefficiencies.
• Sometimes, changing the static nature of the game can be a solution as well.
• Move sequentially; Play repeatedly. These kinds of games have their own kind of analysis!
• When looking at sequential move games, the strategy sets will change!
• Using the Nash Equilibria concept might not be sufficient to get “intuitive” equilibria.
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Readings
• Varian Chapter 29
• Varian Chapter 30 (optional)
• Osborne, An Introduction to Game Theory (optional): A game theory reference
textbook
• Osborne and Rubinstein, A Course in Game Theory (optional): A game theory
reference textbook
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