COU 5: Nash Equilibria with More than Two Persons

In this COU, you’ll explore the application of Nash Equilibrium to games with more than two persons by working with the model we introduced in Lesson 7, COU 7.

Imagine a single election that will determine who occupies S seats in government, where S is an integer greater than or equal to 1. For instance, this could be an election that determines who will hold the S seats representing a particular region in a national legislature. Suppose that K “potential candidates”, where K is an integer greater than S, each choose between two actions:

Suppose that once each of the K persons selects her action, the election is held. The rules of the election and the behavior of the voters under those rules (which we won’t depict in this model!) are such that no matter which of the K persons enter the race, each has the same marginal probability of winning one of the S seats. Specifically, if N of the K persons enter the race, then the marginal probability that any one of them wins one of the S seats is: \left\{ \begin{array}{cl} \frac{S}{N} & \text{if $N \geq S$} \\ 1 & \text{if $N < S$} \end{array} \right\} Finally, suppose that each potential candidate’s utility level is given by: \left\{ \begin{array}{cl} 1 & \text{If she entered the race and won a seat;} \\ 0 & \text{If she stayed out of the race;} \\ -1 & \text{If she entered the race and did not win a seat.} \end{array} \right\}

Prompt 1

Take any one of the potential candidates. Suppose that she expects that a number Q of the potential candidates other than herself will enter the race, where Q is a number greater than or equal to S-1. Write the expression for her expected utility from entering the race. (This will be a function of Q and the number of seats S.)

Prompt 2

Take any one of the potential candidates. Suppose that she expects that a number Q of the potential candidates other than herself will enter the race, where Q is any non-negative integer. Write the expression for her expected utility from staying out of the race. (This is just a number, not a function of any of the parameters.)

Prompt 3

Take any one of the potential candidates. Suppose she expects that a number Q of the potential candidates other than herself will enter the race, where Q is a number greater than or equal to S-1. Say that an action (either “enter the race” or “stay out”) is a best response to that expectation if that action achieves her highest possible expected utility when Q of the potential candidates other than herself enter the race. Using your answers to Prompt 1 and Prompt 2, write the inequality that defines the conditions under which entering the race is a best response to the expectation that Q of the potential candidates other than herself will enter the race. Then re-arrange the inequality so that Q is by itself on one side of the inequality.

Prompt 4

Take any one of the potential candidates. Suppose she expects that a number Q of the potential candidates other than herself will enter the race, where Q is a number greater than or equal to S-1. Continue say that an action (either “enter the race” or “stay out”) is a best response to that expectation if that action achieves her highest possible expected utility when Q of the potential candidates other than herself enter the race. Using your answers to Prompt 1 and Prompt 2, write the inequality that defines the conditions under which staying out of the race is a best response to the expectation that Q of the potential candidates other than herself will enter the race. Then re-arrange the inequality so that Q is by itself on one side of the inequality.

Prompt 5

Imagine a profile of actions in this game in which some number C of the K potential candidates enter the race, and the remaining K-C potential candidates stay out, where C is a number greater than or equal to S and less than the number K of potential candidates. Say that any such profile is a Nash Equilibrium if both of the following are true:

  1. For any potential candidate, entering the race is a best response to the expectation that C-1 of the other potential candidates will enter the race.
  2. For each potential candidate, staying out of the race is a best response to the expectation that C of the other potential candidates will enter the race.

Applying your answers to Prompts 2 and 3, do each of the following:

  1. Taking as given a profile in which C candidates enter the race, where C is a number greater than or equal to S, write the inequality that must hold for part (i) of the definition above to be true, arranging the terms of the inequality so that C is by itself on one side.
  2. Taking as given a profile in which C candidates enter the race, where C is a number greater than or equal to S, write the inequality that must hold for part (ii) of the definition above to be true, arranging the terms of the inequality so that C is by itself on one side.

Prompt 6

Imagine a profile of actions in this game in which some number C of the K potential candidates enter the race, and the remaining K-C potential candidates stay out, where C is a number greater than or equal to S and less than the number K of potential candidates. If you correctly answered parts (a) and (b) of Prompt 4, you know that this profile is a Nash Equilibrium if and only if C is equal to either of two functions of the number of seats S. Write these two functions.

Prompt 7

If you correctly answered Prompt 5, you know that the number of potential candidates who enter the race in any profile in this model that is a Nash Equilibrium in which at least S persons enter the race is equal to either of two functions of the number of seats S. Using the graph below, draw the graph of each of these two functions, as S ranges from 1 to 10 along the horizontal axis…

…Then describe in complete and coherent sentences how an increase in the number of seats S in the election changes the number of candidates that enter the race in any given profile that is a Nash Equilibrium.

Prompt 8

A long-standing hypothesis among political scientists and other observers of representative democracy is that whenever an election is used to decide who will occupy a single office, and in that election the office goes to the candidate who gets the most votes, there will tend to be exactly two viable candidates that run in each election. Is behavior in the Nash Equilibrium profiles in this model consistent with that hypothesis? Answer “yes” or “no” and explain your answer in complete and coherent sentences, using your answer to Prompt 5 in your explanation.