1. What is a degenerate game?
2. Is the following game degenerate or not, justify.
3. Find all nash equilibria in the game in pure or mixed strategies
a b c d
T 4,5 1,4 2,0 1,2
B 6,0 0,1 1,3 0,2
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1. A degenerate game is a type of game theory that refers to a situation where there are multiple Nash equilibria, but one or more of the equilibria are not well-defined or have limited strategic choices.
2. To determine if a game is degenerate or not, we need to analyze the given game. From the table you provided, we can see that there are four strategies for each player, labeled as "a," "b," "c," and "d."
To determine if the game is degenerate or not, we will examine if any of the strategies allow a player to have multiple optimal responses regardless of the other player's strategy. In other words, we are looking for situations where a player can be indifferent between choosing between two or more strategies.
To find any such strategies, we need to check if there are any rows or columns in which the players have the same payoffs for different strategies.
Looking at the table, we see that there are no rows or columns where the payoffs are the same for different strategies. Therefore, there are no multiple optimal responses, and the game is not degenerate.
3. To find all Nash equilibria in the game, we need to analyze each player's best response to the other player's strategies.
Starting with Player 1 (T), we can see that their best response depends on Player 2's strategy. If Player 2 plays "a," Player 1's best response is to choose "T" (4,5). If Player 2 plays "b," Player 1's best response is also "T" (1,4). If Player 2 plays "c," Player 1's best response is "B" (6,0). If Player 2 plays "d," Player 1's best response is "B" (1,2).
Moving on to Player 2 (B), we can see that their best response also depends on Player 1's strategy. If Player 1 plays "T," Player 2's best response is to choose "T" (6,0). If Player 1 plays "B," Player 2's best response is also "B" (0,1).
To find the Nash equilibria, we need to find the strategies that are mutual best responses for both players. In this case, the Nash equilibria occur when both players choose the same strategies, i.e., (T, T) and (B, B).
Therefore, the Nash equilibria in this game are (T, T) and (B, B), where both players choose either (T, T) or (B, B). These equilibria can be in pure strategies (both players choose the same options) or mixed strategies (players choose options probabilistically).