Elaine, George, and Newman are playing the same restaurant game as in Example 10.9. Suppose, however, that they have payoffs as follows: (i) If George and Newman agree, regardless of what Elaine calls out, they then go to the restaurant that George and Newman chose, and Elaine has to pay them three dollars each for coffee. (ii) If Elaine agrees with either George or Newman, but not both, they then go to the restaurant chosen by the two who agreed, and the third person pays 10 dollars to each of the others for dinner. (iii) If all three agree, they then go to the restaurant that they all chose, and George and Newman each pay Elaine five dollars toward her dinner. Find all Nash equilibria for this game

More info: 10.9

EXAMPLE 10.9. Suppose Elaine, George, and Newman are deciding where to go for dinner. They can choose either the Happy Star Chinese Restaurant or the New Yorker Diner. All three agree to simultaneously yell out either "Chinese" or "diner." If two of them agree and the third does not, they all go to the restaurant chosen by the two who agreed, and the third has to pay 10 dollars to each of the others for their dinner. If all three agree, they go to the restaurant that they all chose, and everyone pays for their own dinner. Elaine's payoff array then has entries of the form Eijk = 20 if i does not equal j, j=k , and 0 if i = j = k, and 10 otherwise. The payoff arrays for George and Newman are defined similarly. Let us assume that Elaine, George, and Newman are equally likely to pick the Chinese restaurant or to pick the diner on any given evening. Then their mixed strategies are all the same, namely p = q = r = From this, we can compute the expected value of Elaine's payoff. We find that E = -20(1/4) +0(1/4) + 10(1/2) = 0. Similarly, the expected values for George and Newman are also both 0.

Also see book chapter 10.4 file:///C:/Users/TO/Downloads/Introduction%20to%20Topology%20Pure%20and%20Applied%20by%20Colin%20Adams,%20Robert%20Franzosa%20(z-lib.org).pdf