Finding the mixed-equilibrium strategies for zero-sum games can yield results that seem perplexing at first. In the children's game Rock-Paper-Scissors the optimal mixed strategy is for each player to randomly choose each option one-third of the time. But suppose that when rock beats scissors, the winning player scores 2 points, not 1. How would you expect optimal play to change? Interestingly, the players play rock less, not more. Paper is played one-half of the time, and rock and scissors both drop to one-fourth. Why? Create a payoff matrix and use it to verify that the strategies specified result in a mixed-strategy equilibrium for the modified Rock-Scissors-Paper game. Explain the reasons why rock is played less, not more, using game theory concepts. (250words)
Consider the game of two people approaching one another on a sidewalk. Each chooses right or left. If they make the same choice, they pass one another without a problem and each gets a payoff of 1. If they make opposite choices, they both get payoffs of 0. Find the three Nash equilibria of the game. (One of them is a mixed equilibrium.) Show that the payoff from the mixed equilibrium is only half as good for either player as either of the two pure equilibria.
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