Game Theory Lecture 4 Game Theory Lecture 4

An Introduction to Game theory problem set 4 from M. Osborne’s An Introduction to Game theory To view the problem set Click here →

3 , 3 0 , 6 6 , 0 1 , 1 C Prisoners’ Dilemma C cooperate Some (classical) examples of simultaneous games Prisoners’ Dilemma C Cooperate D defect C cooperate 3 , 3 0 , 6 6 , 0 1 , 1 The ‘D strategy strictly dominates the C strategy

G1(s1,t) > G1(s2,t) G1(s1,t) ≥ G1(s2,t) G1(s1,t) > G1(s2,t) Strategy s1 strictly dominates strategy s2 if for all strategies t of the other player G1(s1,t) > G1(s2,t) Strategy s1 weakly dominates strategy s2 if for all strategies t of the other player G1(s1,t) ≥ G1(s2,t) and for some t G1(s1,t) > G1(s2,t) weakly

Successive deletion of dominated strategies example of strict dominance X 1 , 5 2 , 3 7 , 4 3 , 3 4 , 7 5 , 2 X Nash Equilibrium

??? weak dominance an example 1 , 0 1 , 4 1 , 2 0 , 3 2 , 3 X ? ?

Successive deletion of dominated strategies And Sub-game Perfectness Example 2 1 0 , 3 1 , 4 1 , 2 1 , 0 2 , 3 1 l r 2 1 2

rr lr rl ll 1 , 0 1 , 4 1 , 2 0 , 3 2 , 3 rr lr rl ll 2 1 0 , 3 1 , 4 1 , 2 1 , 0 2 , 3 r l

rr lr rl ll rr lr rl ll ( r,l ) ( l,l ) 1 , 0 1 , 4 1 , 2 0 , 3 2 , 3 Sub-game perfect equiibrium 0 , 3 1 , 2 1 , 4 1 2 ( r,l ) ( l,l ) 2 , 3 1 , 0

rr lr rl ll rr lr rl ll rr lr rl ll rr lr rl ll rr lr rl ll X X X X X 1 , 0 1 , 4 1 , 2 0 , 3 2 , 3 rr lr rl ll 1 , 0 1 , 4 1 , 2 0 , 3 2 , 3 rr lr rl ll 1 , 0 1 , 4 1 , 2 0 , 3 2 , 3 rr lr rl ll 1 , 0 1 , 4 1 , 2 0 , 3 2 , 3 rr lr rl ll 1 , 0 1 , 4 1 , 2 0 , 3 2 , 3 X X weakly dominating 1 1 l r 2 2 2 1 X X X 0 , 3 1 , 2 1 , 4 1 2 ( x , l ) ( x , l ) 2 , 3 1 , 0 delete ( x , r ) delete ( x , r )

Another example of a simultaneous game The Stag Hunt Stag Hare 2 , 2 0 , 1 1 , 0 1 , 1 A generalization to n person game: There are n types of stocks. Stock of type k yields payoff k if at least k individuals chose it, otherwise it yields 0.

Another example of a simultaneous game The Stag Hunt Stag Hare 2 , 2 0 , 1 1 , 0 1 , 1 Stag Hare 2 , 2 0 , 1 1 , 0 1 , 1 Equilibria payoff dominant equilibrium risk dominant equilibrium

Change of payoffs Stag Hare 2 , 2 0 , 1.99 1.99 , 0 1.99 ,1.99

Bach or Stravinsky (BOS) Yet another example of a simultaneous game Battle of the sexes Bach or Stravinsky (BOS) Ballet Boxing 2 , 1 0 , 0 1 , 2 Ballet Boxing 2 , 1 0 , 0 1 , 2 woman man Equilibria

Battle of the sexes woman man 2 , 1 0 , 0 1 , 2 Yet another example of a simultaneous game Battle of the sexes Ballet Boxing 2 , 1 0 , 0 1 , 2 woman man A generalization to a bargaining situation

Nash Demand Game Two players divide a Dollar. Each demands an amount ≥ 0. Each receives his demand if the total amount demanded is ≤ 1. Otherwise they both get 0. Demand of player 2 Demand of player 1 1

Nash Demand Game equilibria a continuum of equilibria Demand of player 2 Demand of player 1 1

Matching Pennies 1 , -1 -1 , 1 last example of a simultaneous game head tails 1 , -1 -1 , 1 no pure strategies equilibrium exists

Matching Pennies 1 , -1 -1 , 1 last example of a simultaneous game head tails 1 , -1 -1 , 1 no pure strategies equilibrium exists

Mixed strategies Matching Pennies 1 , -1 -1 , 1 last example of a simultaneous game Matching Pennies Mixed strategies head tails 1 , -1 -1 , 1 A player may choose head with probability  and tails with probability 1-  no pure strategies equilibrium exists

his payoff is the lottery: last example of a simultaneous game Mixed strategies Matching Pennies 1- head  tails 1 , -1 -1 , 1 player 2 mixes: head tails 1 , -1 -1 , 1 if player 1 plays ‘head’ his payoff is the lottery: if the payoffs are in terms of his vN-M utility then his utility from the lottery is

 1-  Mixed strategies Matching Pennies 1 , -1 -1 , 1 1 , -1 -1 , 1 last example of a simultaneous game Mixed strategies Matching Pennies 1-  head  tails 1 , -1 -1 , 1 player 2 mixes: head tails 1 , -1 -1 , 1 Similarly, if player 1 plays ‘tails’ his payoff is …….

When β = 0.5 player 1 is indifferent between the two strategies last example of a simultaneous game Mixed strategies Matching Pennies 1-  head  tails 1 , -1 -1 , 1 tails player 2 mixes: head tails 1 , -1 -1 , 1 He prefers to play ‘head’ if: He prefers to play ‘tails’ if: When β = 0.5 player 1 is indifferent between the two strategies and any mix of the two

β (1-α , α) α Player 1 prefers to play ‘head’ if: Player 2’s Best Response function ?? Player 1 prefers to play ‘tails’ if: Player 1 is indifferent when player 2’s mix 1 Player 1’s Best Response function β (1-α , α) player 1’s mix tails head α 1

β α When player 1 plays ‘head’ often Player 2 prefers to play ‘tails’ Player 2’s Best Response function ?? Player 2’s Best Response function Nash equilibium α = β= 0 player 2’s mix 1 Player 1’s Best Response function β player 1’s mix α 1

1/2 head 1/2 tails 1 , -1 -1 , 1 tails

Exercises from M. Osborne’s An Introduction to Game Theory EXERCISE 30.1 (Variants of the Stag Hunt) Consider variants of the n-hunter Stag Hunt in which only m hunters, with 2 ≤ m < n, need to pursue the stag in order to catch it. (Continue to assume that there is a single stag.) Assume that a captured stag is shared only by the hunters who catch it. Under each of the following assumptions on the hunters’ preferences, find the Nash equilibria of the strategic game that models the situation. As before, each hunter prefers the fraction 1 / n of the stag to a hare Each hunter prefers the fraction 1 / k of the stag to a hare, but prefers a hare to any smaller fraction of the stag, where k is an integer with m ≤ k ≤ n. The following more difficult exercise enriches the hunters’ choices in the Stag Hunt. This extended game has been proposed as a model that captures Keynes’ basic insight about the possibility of multiple economic equilibria, some of which are undesirable (Bryant 1983, 1994). Next exercise

EXERCISE 31.1 (Extension of the Stag Hunt) Extend the n-hunter Stag Hunt by giving each hunter K (a positive integer) units of effort, which she can allocate between pursuing the stag and catching hares. Denote the effort hunter i devotes to pursuing the stag by ei , a nonnegative integer equal to at most K. The chance that the stag is caught depends on the smallest of all the hunters’ efforts, denoted minj ej. (“A chain is as strong as its weakest link.”) Hunter i’s payoff to the action profile (e1 . . ., en ) is 2minjej -ei . (She is better off the more likely the stag is caught, and worse off the more effort she devotes to pursuing the stag, which means the catches fewer hares.) Is the action profile (e, . . . e), in which every hunter devotes the same effort to pursuing the stag, a Nash equilibrium for any value of e? (What is a player’s payoff to this profile? What is her payoff if she deviates to a lower or higher effort level?) Is any action profile in which not all the players’ effort levels are the same a Nash equilibrium? (Consider a player whose effort exceeds the minimum effort level of all players. What happens to her payoff if the reduces her effort level to the minimum?) Next exercise

Next exercise 2.7.5 Hawk-Dove The Game in the next exercise captures a basic feature of animal conflict. EXERCISE 31.2 (Hawk-Dove) Two animals are fighting over some prey. Each can be passive or aggressive. Each prefers to be aggressive if its opponent is passive, and passive if its opponent is aggressive; given its own stance, it prefers the outcome in which its opponent is passive to that in which its opponent is aggressive. Formulate this situation as a strategic game and find its Nash equilibria. Next exercise

EXERCISE 34.1 (Guessing two-thirds of the average) Each of three people announces an integer from 1 to K. If the three integers are different, the person whose integer is closest to 2/3 of the average of the three integers wins $1. If two or more integers are the same , $1 is split equally between the people whose integer is closest to 2/3 of the average integer. Is there any integer k such that the action profile (k,k,k), in which every person announces the same integer k, is a Nash equilibrium? (If k ≥ 2, what happens if a person announces a smaller number?) Is any other action profile a Nash equilibrium? (What is the payoff of a person whose number is the highest of the three? Can she increase this payoff by announcing a different number?) Last excercise

To return to the presentation click here Game theory is used widely in political science, especially in the study of elections. The game in the following exercise explores citizens’ costly decisions to vote. EXERCISE 34.2 (Voter participation) Two candidates , A and B, compete in an election. Of the n citizen, k support candidate A and m (= n - k) support candidate B. Each citizen decides whether to vote, at a cost, for the candidate she supports, or to abstain. A citizen who abstains receives the payoff of 2 if the candidate she supports wins, 1 if this candidate ties for first place , and 0 if this candidate loses. A citizen who votes receives the payoffs 2 - c, 1 - c, and -c in these three cases, where 0 < c < 1. For k = m = 1, is the game the same (except for the names of the actions) as any considered so far in this chapter? For k = m, find the set of Nash equilibria. (Is the action profile in which everyone votes a Nash equilibrium? Is there any Nash equilibrium in which one of the candidates wins by one vote? Is there any Nash equilibrium in which one of the candidates wins by two or more votes?) What is the set of Nash equilibria for k < m? To return to the presentation click here