There are two workers, A and B, and both workers can choose either work or rest. If both choose to work, they will get 10 individually. Meanwhile, if they both decide to take a rest, they will both obtain 10 as well. However, if A decide to work and B choose to take a rest, A can get 40 yet B gets 20. By contrast, if A choose to take a rest and B decide to keep working, A gets 20 but B gets 30. In the mixed strategy equilibrium, what is the probability that A choose to work?
Question
There are two workers, A and B, and both workers can choose either work or rest. If both choose to work, they will get 10 individually. Meanwhile, if they both decide to take a rest, they will both obtain 10 as well. However, if A decide to work and B choose to take a rest, A can get 40 yet B gets 20. By contrast, if A choose to take a rest and B decide to keep working, A gets 20 but B gets 30. In the mixed strategy equilibrium, what is the probability that A choose to work?
Solution
To find the mixed strategy Nash equilibrium, we need to make each player indifferent to the strategies of the other player. This means that the expected payoff from choosing to work should be equal to the expected payoff from choosing to rest for both players.
Let's denote the probability that worker A chooses to work as p and the probability that worker B chooses to work as q.
The expected payoff for worker A when they choose to work is: 10q + 40(1-q) The expected payoff for worker A when they choose to rest is: 20q + 10(1-q)
Setting these two equal to each other gives us:
10q + 40(1-q) = 20q + 10(1-q)
Solving this equation for q gives us q = 1/2.
So, in the mixed strategy Nash equilibrium, the probability that worker A chooses to work (p) is 1/2.
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