Mixed strategy equilibria

• Mixed strategy: a player chooses between the moves according to a probability distribution

• Suppose each player chooses S with probability 1/10. Is that a Nash equilibrium?

• Consider payoffs to P1 while keeping P2’s strategy fixed– The payoff of P1 choosing S is (1/10)(–10) + (9/10)1 = –1/10– The payoff of P1 choosing C is (1/10)(–1) + (9/10)0 = –1/10– Is there a different strategy that can improve P1’s payoff?– Similar reasoning applies to P2

S CS -10, -10 -1, 1C 1, -1 0, 0

Straight

Chicken Straight

ChickenPlayer 1 Player 2

Finding mixed strategy equilibria

• Expected payoffs for P1 given P2’s strategy: P1 chooses S: q(–10) +(1–q)1 = –11q + 1P1 chooses C: q(–1) + (1–q)0 = –q

• In order for player P1 to be willing to randomize, the expected payoffs of the two actions should be the same–11q + 1 = –q or q = 1/10Similarly, p = 1/10

P1: Choose Swith prob. p

P1: Choose Cwith prob. 1-p

P2: Choose S with prob. q -10, -10 -1, 1

P2: Choose C with prob. 1-q 1, -1 0, 0

Mixed strategy equilibria: Another example

• Pure strategy equilibria:– (Ballet, ballet) or (football, football)

Wife:Ballet

Wife:Football

Husband:Ballet 3, 2 0, 0

Husband:Football 0, 0 2, 3

Mixed strategy equilibria: Another example

• Payoff to wife assuming she chooses ballet: 3q• Payoff to wife assuming she chooses football: 2(1–q)• 3q = 2(1–q) or q = 2/5• Payoff to husband assuming he chooses ballet: 2p• Payoff to husband assuming he chooses football: 3(1-p)• 2p = 3(1–p) or p = 3/5• Mixed strategy equilibrium: wife picks ballet w/ probability 3/5 and

husband picks football with probability 3/5

Wife:Ballet w/ prob. p

Wife:Football w/ prob. 1-p

Husband:Ballet w/ prob. q 3, 2 0, 0

Husband: Football w/ prob 1-q 0, 0 2, 3

Mixed strategy equilibria: Another example

• Mixed strategy equilibrium: wife picks ballet w/ probability 3/5 and husband picks football with probability 3/5

• How often do they end up in different places?P(wife = ballet, husband = football or wife = football, husband = ballet)

= 3/5 * 3/5 + 2/5 * 2/5 = 13/25• What is the expected payoff for each?

3/5 * 2/5 * 3 + 3/5 * 2/5 * 2 = 6/5• What is the payoff for always going to the other’s preferred event?

Wife:Ballet w/ prob. p

Wife:Football w/ prob. 1-p

Husband:Ballet w/ prob. q 3, 2 0, 0

Husband: Football w/ prob 1-q 0, 0 2, 3

Back to rock-paper-scissors• Zero-sum game: want to find

minimax solution for P1

P2

P1

0,0 1,-1 -1,1

-1,1 0,0 1,-1

1,-1 -1,1 0,0

Back to rock-paper-scissors• Zero-sum game: want to find

minimax solution for P1• Let r, p, s = probability of P1

playing rock, paper, scissors • Let’s find the expected payoffs for P1

given different actions of P2:P2 plays rock: u = r(0) + p(1) + s(–1) = p – s P2 plays paper: u = r(–1) + p(0) + s(1) = s – r P2 plays scissors: u = r(1) + p(–1) + s(0) = r – p

• P2 is trying to minimize P1’s utility, so we haveu p – s; u s – r; u r – p

• To find r, p, s, maximize u subject to the above constraints and r + p + s = 1– Linear programming problem– Solution is (1/3, 1/3, 1/3)

P2

P1

0 1 -1

-1 0 1

1 -1 0

Computing Nash equilibria• Any game with a finite set of actions has at least

one Nash equilibrium• If a player has a dominant strategy, there exists a

Nash equilibrium in which the player plays that strategy and the other player plays the best response to that strategy

• If both players have strictly dominant strategies, there exists a Nash equilibrium in which they play those strategies

Computing Nash equilibria• For a two-player zero-sum game, simple linear

programming problem• For non-zero-sum games, the algorithm has worst-

case running time that is exponential in the number of actions

• For more than two players, and for sequential games, things get pretty hairy

Nash equilibria and rational decisions

• If a game has a unique Nash equilibrium, it will be adopted if each player– is rational and the payoff matrix is accurate– doesn’t make mistakes in execution– is capable of computing the Nash equilibrium – believes that a deviation in strategy on their part will

not cause the other players to deviate– knows that everybody knows that everybody meets

these conditions (common knowledge)

Nash equilibria and rational decisions

Do you have a dominant strategy?

Play dominant strategy

yes no

Do you know what the opponent will do?

yes no

Maximize utility Is opponent rational?

yes no

Can agree on a Nashequilibrium?

yes no

Play the equilibrium strategy

Maximize worst-case outcome

Maximize worst-case outcome

More fun stuff: Ultimatum game• Alice and Bob are given a sum of money S to divide

– Alice picks A, the amount she wants to keep for herself– Bob picks B, the smallest amount of money he is willing to accept– If S – A B, Alice gets A and Bob gets S - A– If S – A < B, both players get nothing

• What is the Nash equilibrium?– Alice offers Bob the smallest amount of money he will accept:

S – A = B – Alice offers Bob nothing and Bob is not willing to accept anything

less than the full amount: A = S, B = S (both players get nothing)• How would humans behave in this game?

– If Bob perceives Alice’s offer as unfair, Bob will be likely to refuse– Is this rational?

• Maybe Bob gets some positive utility for retaliating against Alice?

Repeated games• What if the Prisoner’s Dilemma

is played for many rounds and the players remember what happened in the previous rounds?

• What if the number of games is fixed and known in advance to both players?– Then the equilibrium is still to defect

• If the number of games is random or unknown, cooperation may become a equilibrium strategy– Perpetual punishment: cooperate unless the other player has

ever defected– Tit for tat: start by cooperating, repeat the other player’s previous

move for all subsequent rounds• In order for these strategies to work, the players must

know that they have both adopted them

Cooperate Defect

Cooperate -1,-1 0,-10

Defect -10,0 -5,-5

Some multi-player games• The diner’s dilemma

– A group of people go out to eat and agree to split the bill equally. Each has a choice of ordering a cheap dish or an expensive dish (the utility of the expensive dish is higher than that of the cheap dish, but not enough for you to want to pay the difference)

– Nash equilibrium is for everybody to get the expensive dish• El Farol bar problem (W. Brian Arthur)

– If less than 60% of the town’s population go to the bar, they will have a better time than if they stayed at home. If more than 60% of the people go to the bar, the bar will be too crowded and they will have a worse time than if they stayed at home

– Nash equilibrium must be a mixed strategy

Mechanism design (inverse game theory)

• Assuming that agents pick rational strategies, how should we design the game to achieve a socially desirable outcome?

• We have multiple agents and a center that collects their choices and determines the outcome

Auctions• Goals

– Maximize revenue to the seller– Efficiency: make sure the buyer who values the goods

the most gets them– Minimize transaction costs for buyer and sellers

Ascending-bid auction• What’s the optimal strategy for a buyer?

– Bid until the current bid value exceeds your private value• Usually revenue-maximizing and efficient, unless

the reserve price is set too low or too high• Disadvantages

– Collusion– Lack of competition– Has high communication costs

Sealed-bid auction• Each buyer makes a single bid and communicates it to the

auctioneer, but not to the other bidders– Simpler communication– More complicated decision-making: the strategy of a buyer depends on

what they believe about the other buyers– Not necessarily efficient

• Sealed-bid second-price auction: the winner pays the price of the second-highest bid– Let V be your private value and B be the highest bid by any other buyer– If V > B, you can win by bidding above B – in particular, bid V– If V < B, your optimal strategy is to bid below B – in particular, bid V– Therefore, your dominant strategy is to bid V– This is a truth revealing mechanism

Dollar auction• A dollar bill is being auctioned off. It goes to the highest

bidder, but the second-highest bidder also has to pay– Player 1 bids 1 cent– Player 2 bids 2 cents– …– Player 2 bids 98 cents– Player 1 bids 99 cents

• If Player 2 passes, he loses 98 cents, if he bids $1, he might still come out even– So Player 2 bids $1

• Now, if Player 1 passes, he loses 99 cents, if he bids $1.01, he only loses 1 cent– …

• What went wrong?– When figuring out the expected utility of a bid, a rational player

should take into account the future course of the game• How about Player 1 starts by bidding 99 cents?

Distribution of goods• A city decides to install wireless networks in a few

neighborhoods – Can’t afford to give wireless to every neighborhood that wants it

– should pick neighborhoods that would value it the most– If the city asks each neighborhood how much they value the free

wireless, the neighborhoods will be tempted to lie• Mechanism: each winner would have to pay a fee equal

to the highest reported value among the losers – Winners are happy because fee is less than their utility– Losers are happy because they don’t value wireless more than

the amount of the fee– Truth-revealing: the dominant strategy for each neighborhood

is to report true utility

Tragedy of the commons• States want to set their policies for controlling emissions

– Each state can reduce their emissions at a cost of -10 or continue to pollute at a cost of -5

– If a state decides to pollute, -1 is added to the utility of every other state

• What is the dominant strategy for each state?– Continue to pollute– Each state incurs cost of -5-49 = -54– If they all decided to deal with emissions, they would incur a cost

of only -10 each• Mechanism for fixing the problem:

– Tax each state by the total amount by which they reduce the global utility (externality cost)

– This way, continuing to pollute would now cost -54

Game theory issues• Is it applicable to real life?

– Humans are not always rational– Utilities may not always be known– Other assumptions made by the game-theoretic

model may not hold– Political difficulties may prevent theoretically optimal

mechanisms from being implemented• Could it be more applicable to AI than to real life?

– Computing Nash equilibria in complicated games is difficult

– Relationship between Nash equilibrium and rational decision making is subtle