Question

Drug dealers and arch rivals Carl and Jason are both armed and find themselves unexpectedly in a dangerous stand-off. In an attempt to protect themselves, each of them has to choose from one of two actions: attempt to shoot the rival (S) or dodge and take cover (D). Each of them must decide simultaneously and independently, with each man aiming to maximize his own chance of survival. They face death not only from the rival’s shooting but also from ran- dom firing by police who are in hot pursuit of them. Their survival rates in each scenario are given as follows: · - If Carl and Jason both choose S, then their survival probabilities are 34% and 29% respectively. · - If Carl and Jason both choose D, then their survival probabilities are 36% and 39% respectively. · - If Carl chooses S while Jason chooses D, then their survival probabilities are 38% and 32% respectively. · - If Carl chooses D while Jason chooses S, then their survival probabilities are 35% and 38% respectively. Based on the above information, answer the following questions:

(d) Suppose now that instead of interacting simultaneously, the players move sequentially with either Carl or Jason moving first and their choice is observed by the player moving second. In particular, then assume that Jason moves first. Represent the extensive form of this sequential move game using a game tree. Predict the outcome of this game. Does the player moving first have an advantage?

Answer 1

The Sequential Game Tree is made as follows

Since Jason moves first, he knows that if he Shoots, Carl will have a better payoff of 35% by taking cover. In that case Jason's probability of survival will be 38%

However, if Jason Takes Cover, Carl will Shoot and get a better survival rate of 38%. Jason will have a survival probability of 32%

SO if Jason shoots, he can get a survival probability of 38% while if he takes cover, his survival probability will be 32% by backward iteration.

So the outcome will be that Jason will prefer to shoot and hence Carl will take cover. The survival probabilities will be 38% for Jason and 35% for Carl.

If Carl moves First, we can see by similar backward iteration that both will Take Cover as the outcome and as a result of this Carl's probability of survival will be 36% up from 35% and Jason's survival probability will be 39% up from 38%. So going first is advantageous for Carl but it is not advantageous for Jason. So the person going first does not necessarily have any advantage.

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