Players A and B are playing a simultaneous moves game and both can choose either strategy S1 or strategy S2. If both choose S1 both receive 0. If both choose S2 both receive -2. If their chosen strategies differ they both receive -4.
(a)Write out a table representing each player¡¦s strategies and payoffs (payoff matrix).
(b)State the set of strategy profiles.
(c)Suppose player A plays S1. What is B¡¦s best response?
(d)Suppose player B plays S2. What is A¡¦s best response?
(e)Does A have a dominant strategy?
(f)Is there a unique outcome by elimination of dominated strategies?
(g)What is (or are) the pure strategy NE to this game?
(h)Explain precisely why it is the NE (if more NE just pick one of them). Pick one strategy profile that is NOT a NE and explain precisely why.
(i)Which of the NE do you find most likely to prevail in the real world. Why? What about if the game was played repeatedly?
(a) The payoff matrix can be represented as follows:
| A: S1 | A: S2
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B: S1 | 0 | -4
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B: S2 | -4 | -2
In the above matrix, the rows represent the choices of player B, and the columns represent the choices of player A. The numbers in the cells represent the payoffs for player A and player B when they choose their respective strategies.
(b) The set of strategy profiles is {S1, S1}, {S2, S2}, {S1, S2}, and {S2, S1}, which are all possible combinations of strategies for player A and player B.
(c) If player A plays S1, player B's best response is also S1. This is because both players receive a payoff of 0 when they both choose S1, which is higher than the payoff of -4 when their strategies differ.
(d) If player B plays S2, player A's best response is also S2. This is because both players receive a payoff of -2 when they both choose S2, which is higher than the payoff of -4 when their strategies differ.
(e) Player A does not have a dominant strategy because there is no strategy that always gives a higher payoff regardless of the other player's choice.
(f) By elimination of dominated strategies, we can see that strategy S2 for both players is dominated. This is because when both players choose S2, they receive a lower payoff compared to when they choose S1. The elimination of dominated strategies leaves us with the strategy profile {S1, S1}.
(g) The pure strategy Nash Equilibrium (NE) to this game is {S1, S1}. This is because both players choose S1, and no player has an incentive to deviate from this strategy. If either player switches to S2, they would receive a lower payoff.
(h) The strategy profile {S2, S2} is not a Nash Equilibrium. If both players choose S2, they both receive a payoff of -2. However, if player A deviates and chooses S1, they can achieve a higher payoff of 0. Similarly, if player B deviates and chooses S1, they can also achieve a higher payoff of 0. Therefore, this strategy profile is not a Nash Equilibrium because at least one player has an incentive to deviate.
(i) In the real world, the most likely NE to prevail would be {S1, S1}. This is because both players choose the strategy that gives them the highest payoff given the other player's strategy. In terms of repeated play, the NE {S1, S1} is also likely to prevail because it is a stable strategy that neither player has an incentive to deviate from. Over time, players learn to choose the strategy that maximizes their payoffs, leading to the convergence towards the NE.