(Tadelis 2.11). You\'re a sprinter, and in practice today you fell and hurt your
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Question
(Tadelis 2.11). You're a sprinter, and in practice today you fell and hurt your leg. An x-ray suggests that it's broken with probability 0.2. Your problem is whether you should participate in next week's tournament. If you run, you think you'll win with probability 0.1. If your leg is broken and you run, then it will be further damaged and your payoffs are as follows:
+100 if you win the race and your leg isn't broken
+50 if you win and your leg is broken
0 if you lose and your leg isn't broken
-50 if you lose and your leg is broken
0 if you don't run and your leg isn't broken
-10 if you don't run and your leg is broken
Suppose now there is a fixed cost to running, c>0. Thus, for example, if you win the race and your leg isn't broken, your payoff is 100-c. Find c such that you are indifferent between running and not running in Question 1.
Explanation / Answer
Let us first find the expected payoff for the race, which we’ll be doing by taking into account all he possibilities that could arise in course of the race.
Let us first list the probabilities of my leg being broken or not and the odds of winning the tournament.
The probability of my leg being broken is 0.2;
Therefore, the probability that my leg not being broken is (1 - 0.2) = 0.8.
The probability of winning the tournament is 0.1;
Therefore, the probability of losing the race is (1 – 0.1) = 0.9.
Let us now list the probabilities for the following situations—
Now if I don’t run, then there is no question of payoffs, rather we’ll be talking about the cost that I have to pay to cure my leg if it’s broken.
If I don’t run and my leg is not broken then I’ll not have to incur any cost;
Whereas if I don’t run and my leg is broken, then I’ll have to incur a cost of 10 units; denoted by (-10) in the payoff possibilities.
Now, let us find the expected payoff in the following way—
Expected Payoff = 100 * 0.08 + 50 * 0.02 + 0 * 0.72 + (-50) * 0.18
= 8 + 1 + 0 – 9
= 9 – 9
= 0
Therefore, we can see that if I took participation into the tournament, then my expected payoff from running the tournament is 0. Also, there is a fixed cost of c > 0 for running the tournament. So, the net payoff from running the tournament is, 0 – c = (- c).
If I do not run the tournament, then the payoff that I’ll be getting is, 0 + (-10) = (- 10). This means I’ll have to incur 10 units of cost in case I skipped the race to cure my broken leg.
Now, if we compare the above two situations of running and not running the tournament, then we can say if the value of ‘c’ becomes 10, that is, if I have to incur a fixed cost of 10 units for taking part into the tournament, then I’ll become indifferent between running and not running the tournament as in both the situations I’ll be incurring the same cost of 10 units expectedly whether as tournament entry fee or fee for curing my leg given that it is broken.
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