ECO316H1 Lecture 10: Lecture 10 Repeated Games and Collusion

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lilacoyster158

5 Apr 2019

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Repeated games: imagine prisoner"s dilemma played repeatedly. Interpret as duopoly, with high price and low price actions. After each game, participant informed of opponent"s action. Stage game: same set of players repeatedly play strategic game, each player observes other players" previous actions, each player can condition action on action profiles in previous periods, play continues indefinitely but players discount future payoffs. Players may be "impatient: prefer consumption now rather than later, game may end with positive probability after any period. Payoffs: payoff function is ui, payoff to sequence a1, a2, a3 of action profiles is. Can be reduced to if payoff same in every period: strategy in repeated game is function that specifies action in each period for every possible. Strategies history of action profiles: strategy is function that specifies: Action in second period for every possible action profile in first period. Action in third period for all possible action profiles in first and second period.

Related Questions

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).

PrisonerÃ¢â¬â¢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 1Ã¢â¬â¢s best response function (choice of p as a function of q), and player 2Ã¢â¬â¢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 ÃÂ± 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 personÃ¢â¬â¢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 (ÃÂ±)			Player 2 is weak (1-ÃÂ±)			Player 2 is strong (ÃÂ±)

After writing all the strategies and payoffs in the same matrix, find the Bayesian Nash equilibriums, depending on the value of ÃÂ± (ÃÂ± Ã¢â°Â¤ 1/2 or ÃÂ± Ã¢â°Â¥1/2).

jadebuffalo465

QUESTION 1

Stackelberg duopoly game is also known as the ________ model. If we change the Stackelberg ______ competition game to a simultaneous-move game, we get the ______ game results.

	A.	leader-follower, quantity; Cournot
	B.	Competitive fringe; price; backward induction
	C.	leader-follower, quantity; Bertrand
	D.	entry, price; Cournot

QUESTION 2

Comparing Stackelberg and Cournot competition results, we can say that the _____ is better off while the ______ is worse off under Stackelberg than under Cournot results. This result show that there is _______________ advantage.

	A.	entrant, incumbent, investment
	B.	leader, follower, first-mover
	C.	follower, leader, a size
	D.	incumbent, entrant, first-mover

QUESTION 3

Mark all the FALSE statements

	A.	An equilibrium is a collection of strategies (and a strategy is a complete plan of action), whereas an outcome describes what will happen only in the contingencies that are expected to arise, not in every contingency that might arise.
	B.	In games of complete but imperfect information, backward induction is still the strongest process to solve the model to get unique equilibrium.
	C.	All subgame perfect Nash equilibria (SPNE) are Nash equilibria (NE), but not all Nash equilibria is SPNE
	D.	We cannot apply the notion of Nash equilibrium to dynamic games of complete information if we allowed a player s strategy to leave unspecified actions in some contingencies.
	E.	A game can be of perfect information whenever Nature or Luck does not play, each information set does not necessarily need to have a single node.

QUESTION 4

True or false. Mark the correct sequence:
I. Simultaneity of moves means that these games have imperfect information.
II. Dynamic games of complete and perfect information do not necessarily need that a player observes all the previous moves, just part of them are fine.
III. Backward induction and subgame perfect equilibrium concept lead to the same result in games of incomplete information.
IV. Any game in extensive form is a subgame itself.

	A.	TFFT
	B.	TTTF
	C.	FTFF
	D.	TTFT

QUESTION 5

The subgame perfect Nash equilibrium is the equilibrium associated with ____________ outcome. Subgame perfect equilibrium ________ non-credible threats.

	A.	Maxmin; involves
	B.	the backward induction; does not involve
	C.	Iterative deletion of weakly dominated strategies; rules out
	D.	the backward induction; includes

ECO316H1 Lecture 10: Lecture 10 Repeated Games and Collusion

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