Problem 1 In each of the three games shown below, let p be the probability that

Question

Problem 1

In each of the three games shown below, let p be the probability that player 1 plays cooperates (and 1- p the probability that player 1 defects), and let q be the probability that Player 2 plays cooperates (and 1- q the probability that player 2 defects).

Prisoner’s Dilemma

Player 2

Player 1

cooperate

defect

cooperate

70,70

10,80

defect

80,10

40,40

Stag Hunt

Player 2

Player 1

cooperate

defect

cooperate

70,70

5,40

defect

40,5

40,40

Chicken

Player 2

Player 1

cooperate

defect

cooperate

70,70

50,80

defect

80,50

40,40

1. For each game, draw a graph with player 1’s best response function (choice of p as a function of q), and player 2’s best response function (choice of q as a function of p), with p on the horizontal axis and q on the vertical axis.

2. Using this graphs, find all the Nash equilibriums for the game, both pure and mixed strategy Nash equilibriums (if any). Label these equilibriums on the corresponding graph.

3. In those games that have multiple pure strategy Nash equilibriums, how do the expected payoffs from playing the mixed strategy Nash equilibrium compare with the payoffs from playing the pure strategy Nash equilibriums? Which type of strategy (mixed or pure) would players prefer to play in these games?

Problem 2

Two people are involved in a dispute. Player 1 does not know whether player 2 is strong or weak; she assigns probability α to player 2 being strong. Player 2 is fully informed. Each player can either fight or yield. Each player obtains a payoff of 0 is she yields (regardless of the other person’s action) and a payoff of 1 if she fights and her opponent yields. If both players fight, then their payoffs are (-1; 1) if player 2 is strong and (1;-1) if player 2 is weak. The Bayesian game is the following, depending on the type of player 2:

Y

F

Y

F

Y

0, 0

0, 1

Y

0, 0

0, 1

F

1, 0

-1, 1

F

1, 0

1, -1

Player 2 is strong (α)

Player 2 is weak (1-α)

Player 2 is strong (α)

After writing all the strategies and payoffs in the same matrix, find the Bayesian Nash equilibriums, depending on the value of α (α ≤ 1/2 or α ≥1/2).

Player 2

Player 1

cooperate

defect

cooperate

70,70

10,80

defect

80,10

40,40

Explanation / Answer

Game theory is a study of strategic decision making. More formally, it is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers."[1] An alternative term suggested "as a more descriptive name for the discipline" is interactive decision theory.[2] Game theory is mainly used in economics, political science, and psychology, as well as logic and biology. The subject first addressed zero-sum games, such that one person's gains exactly equal net losses of the other participant(s). Today, however, game theory applies to a wide range of behavioral relations, and has developed into an umbrella term for the logical side of decision science, to include both human and non-humans, like computers. Classic uses include a sense of balance in numerous games, where each person has found or developed a tactic that cannot successfully better her results, given the strategies of other players. Modern game theory began with the idea regarding the existence of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann. Von Neumann's original proof used Brouwer's fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by his 1944 book Theory of Games and Economic Behavior, with Oskar Morgenstern, which considered cooperative games of several players. The second edition of this book provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty. This theory was developed extensively in the 1950s by many scholars. Game theory was later explicitly applied to biology in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been widely recognized as an important tool in many fields. Eight game-theorists have won the Nobel Memorial Prize in Economic Sciences, and John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology.

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