Mixed Strategies in Game Theory
When playing a game, a player's goal is to maximize their expected payoff. However, when mixed strategies are involved, calculating the expected payoff becomes a bit more complicated.
To calculate the expected payoff for a mixed strategy, we first need to determine the probability distribution of the opponent's strategies. This is because the payoff of each strategy depends on the opponent's strategy.
Once we have the probability distribution, we can calculate the expected payoff for each of our own strategies. We do this by multiplying the probability of the opponent playing a particular strategy by the payoff we would receive if we played our corresponding strategy. We then sum up these values for each of our strategies to get our overall expected payoff.
For example, let's consider the following game:
| | L | R | |---|---|---| | T | 2,1 | 0,0 | | B | 0,0 | 1,2 |
Suppose we play the mixed strategy (p,1-p) where we play T with probability p and B with probability 1-p. If our opponent plays strategy L with probability q and strategy R with probability 1-q, then the probability distribution is (pq, p(1-q), (1-p)q, (1-p)(1-q)).
Now, let's calculate the expected payoff of playing T. If our opponent plays strategy L, we get a payoff of 2 with probability pq and a payoff of 0 with probability (1-p)q. If our opponent plays strategy R, we get a payoff of 1 with probability p(1-q) and a payoff of 0 with probability (1-p)(1-q). So, the expected payoff of playing T is 2pq + p(1-q).
Similarly, we can calculate the expected payoff of playing B. If we add these up, we get our overall expected payoff for the mixed strategy (p,1-p). This process can be repeated for different mixed strategies to find the one with the highest expected payoff.
It's important to note that some games may not have a unique mixed strategy equilibrium, and in those cases, we may have to resort to other methods to find the solution.
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