Question

Player 1 and Player 2 are considering whether to sign a contract or not. The resulting payoffs to each player for the alternative strategies are as follows, where X≥0 and Z≥0 are the non-negative parameters. Player 2 Player 1 Don a) Suppose X=Z=0, and assume that both players make their decisions simultaneously. Which firm has a dominant strategy (if any)? Find the Nash equilibrium for this game (if any). b) Find the values of X for which Sign is the maximin strategy for Player 1. For what values of Z is Don't sign the maximin strategy for Player 2 ? c) Now suppose X=Z=5. As above, the players make their decisions simultaneously, but the game is now repeated. i. If the game is played for 10 periods, what will be the resulting equilibrium? ii. Suppose neither player knows when the game will end, and Player 1 adopts a grim trigger strategy. At what level of the discount rate r will Player 2 the cooperative outcome be achieved? Explain. d) Now suppose X=Z=2, and assume that Player 2 makes its decision before Player 1 . Write down the extensive form of the sequential game. Identify all Nash equilibria (if any) in this game. Does Player 2 have a first-mover advantage? Explain.

Answer 1

In a game with two players making simultaneous moves, dominant strategies and Nash equilibrium are key to predicting outcomes. With repeated plays or sequential moves, strategies may adapt, and additional factors come into play. Maximin strategies and Nash equilibria help assess the choices of rational players.

Step-by-step explanation:

When considering the outcomes of a decision-making scenario involving two players with simultaneous moves, the concept of dominant strategy and Nash equilibrium are crucial elements. The dilemma faced by the players resembles the classical prisoner's dilemma where cooperation or defection can lead to varying payoffs. When X and Z are set to zero, it simplifies the situation, allowing for clear assessment of the possible strategies and likely outcomes. If both players opt for their dominant strategies, assuming rationality, they will arrive at a Nash equilibrium.

For maximin strategies, Player 1 will look at the worst payoffs from each of their strategies and choose the one with the highest payoff, while Player 2 will do the same. If the game is repeated, the strategies may evolve, and factors like the discount rate r become relevant for strategies such as the grim trigger. Understanding the strategic interactions in a sequential game where Player 2 moves first requires the analysis of possible Nash equilibria and the evaluation of a first-mover advantage.