Exercises and Projects for Chapter 2: Simultaneous Games

1. *** a) Write down the matrices of the SIMULTANEOUS LEGISLATORS VOTE game in the variant where each of the three voters has also the option to abstain. The raise only passes if more agree than voting against. The loss of face by abstaining is relatively small, only $200.
   b) Solve that game, using the approaches discussed above.
2. Assume a simultaneous two-player game has the best response digraph shown to the right. Display a possible payoff bimatrix. Can you find a possible zero-sum payoff bimatrix generating this best response digraph?
3. Consider the following two-player game.

   	L	M	R
   U	1,1	3,4	2,1
   M	2,4	2,5	8,1
   D	3,3	0,4	0,9

   • Find the Maximin moves for both players.
   • Which moves are dominated?
   • Find the bimatrix obtained by IESD.
   • Find the bimatrix obtained by IEWD.
   • Draw the best response digraph.
   • Are there any Nash equilibria?
4. Analyze the following six two-person zero-sum games (Maximin moves, domination, best response digraph, Nash equilibria):

   L     R   U   1     2   D   3     4

   L    R   U   1     2   D   4     3

   L     R   U   1     3   D   2     4

   L     R   U   1     3   D   4     2

   L     R   U   1     4   D   2     3

   L     R   U   1     4   D   3     2

5. Consider the 2 person variant of the GUESS THE AVERAGE(2,4) game where every player can just choose one of the numbers 1,2,3,4. Create the payoff bimatrix. Decide whether the game has a dominant move equilibrium, an IEWD equilibrium, an IESD equilibrium, or a Nash equilibrium.
6. *** In the TWO BARS example above, it is obvious that lack of tourists make the situation more competitive. Assume the number of natives is fixed as 4000. For which number of tourists would both bars choose $ 4 as price for the beer? For which tourist numbers is $ 2 possible, and for which tourist numbers is $ 5 possible?
7. Write down the payoff bimatrix of the following game. Find Maximin moves, domination, condensed best response digraph, and all pure Nash equilibria.
   SCHEDULING A DINNER PARTY: Ann and Beth are not on speaking terms, but have a lot of common friends. Both want to invite these friends to a dinner party this weekend. Possible are Friday or Saturday evening. Both slightly prefer Saturday. If both set the party at the same time, this will be considered a disaster with a payoff of -10 for both. If one plans the party on Friday and the other on Saturday, the one having the Saturday party gets a payoff of 5, and the other of 4.
8. Analyze the following game. Create payoff bimatrices consistent to the information given. Explain your choices. Then find the maximin moves, domination, and all pure Nash equilibria.
   SELECTING CLASS: Adam, Bill, and Cindy are registering for one foreign language class independently and simultaneously. The available classes are ITA100 and FRE100. All three are almost indifferent between the two choices, but they are not indifferent with whom to share the class. More precisely, Bill and Cindy want to be in the same class, but want to avoid Adam. On the other hand, Adam wants to be in the same class as Bill and Cindy, or even better, both.
9. DEADLOCK: Two players play a symmetric game where each one can either cooperate or defect. If both cooperate, both get an payoff of 1. If both defect both get a payoff of 2. If one cooperate but the other defects, the one cooperating gets a payoff of 0, and the other defecting a payoff of 3.
   Draw the bimatrix of the game. Find the Maximin moves, possible domination, draw the best response digraph, and find all pure Nash equilibria.
10. STAG HUNT: Two players play a symmetric game where each one can either hunt stag or hare. If both hunt stag, both get an payoff of 3. If both hunt hare, both get a payoff of 1. If one hunts stag and the other hare, the stag hunter gets a payoff of 0, and the hare hunter a payoff of 2.
    Draw the bimatrix of the game. Find the Maximin moves, possible domination, draw the best response digraph, and find all pure Nash equilibria.
11. CHICKEN: Two players play a symmetric game where each one can either play Dove or Hawk. If both play Dove, both get an payoff of 2. If both play hawk, both get a payoff of 0. If one plays Dove and the other Hawk, the one playing Dove gets a payoff of 1, and the other one a payoff of 3.
    Draw the bimatrix of the game. Find the Maximin moves, possible domination, draw the best response digraph, and find all pure Nash equilibria.
12. BULLY: Two players play the following symmetric game

    	Cooperate	Defect
    Cooperate	2, 1	1, 3
    Defect	3, 0	0, 2

    (compare [Poundstone 1993]).

    Draw the bimatrix of the game. Find the Maximin moves, possible domination, draw the best response digraph, and find all pure Nash equilibria.
13. Two cars are meeting at an intersection and want to proceed as indicated by the arrows in the picture. Each player can proceed or yield. If both proceed, there is an accident. A would have a payoff of -100 in this case, and B a payoff of -1000 (since B would be made responsible for the accident, since A has the right of way). If one yields and the other proceeds, the one yielding has a payoff of -5, and the other one of 5. If both yield, it takes a little longer until they can proceed, so both have a payoff of -10. Analyze this simultaneous game, draw the payoff bimatrix and find pure Nash equilibria.
14. Three cars are meeting at an intersection and want to proceed as indicated by the arrows in the picture. Each player can proceed or yield. If two with intersecting paths proceed, there is an accident. The one having the right of way has a payoff of -100 in this case, the other one a payoff of -1000. If a car proceeds without causing an accident, the payoff for that car is 5. If a car yields and the all others intersecting its path proceed, the yielding car has a payoff of -5. If a car yields and a conflicting path car as well, it takes a little longer until they can proceed, so both have a payoff of -10. Analyze this simultaneous game, draw the payoff bimatrices and find all pure Nash equilibria.
15. Solve the SIMULTANEOUS ULTIMATUM GAME: Display the payoff bimatrix, and investigate Maximin moves, domination, best responses, and whether there are any equilibria.
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1. Project: SELECTING CLASS: Adam, Bill, and Cindy are registering for one foreign language class independently and simultaneously. The available classes are ITA100 and FRE100. All three are almost indifferent between the two choices, but they are not indifferent with whom to share the class. More precisely, Bill and Cindy want to be in the same class, but want to avoid Adam. On the other hand, Adam wants to be in the same class as Bill and Cindy, or even better, both.
   Assume that the payoff for each player is the sum of the payoffs of having the other two in the same class or not, and the payoff for being in the right class. Assume that the payoff for being in the right class is 0.1 or -0.1, and the payoff for having the other in the same class are 1 or -1. There are 8 possible cases. In which one do we get Nash equilibria, and in which one of them do we get just one Nash equilibrium?
2. Project*: GENERALIZED SELECTING CLASS: Adam, Bill, and Cindy are registering for one foreign language class independently and simultaneously. The available classes are ITA100 and FRE100. All three are almost indifferent between the two choices, but they are not indifferent with whom to share the class.
   Assume that the payoff for each player is the sum of the payoffs of having the other two in the same class or not, and the payoff for being in the right class. Assume that the payoff for being in the right class is 0.1 or -0.1, and the payoff for having the other in the same class are 1, 0, or -1. There are many cases. Can you classify in which cases we get Nash equilibria, and in which one of them do we get just one Nash equilibrium?
   What can be said about the stability of the Nash equilibria obtained? What if these payoffs are not exactly 1, 0, -1, 0.1, -0.1, but about that much?
3. Project*: ELECTION, Part 1:

   Ann and Beth are running for president of the USB. There are three states. Similar to the president of the USA, this president is elected by votes from the three states, where states have different numbers of votes, and cannot split their votes. A state votes for that candidate that visited the state more often. In case of a tie, they abstain. We assume that the payoff for the winner is 1, and the payoff for the loser is -1. In case of a tie the payoffs are 0 for both.

   It's three days before the election, and the two candidates have to determine now, simultaneously, which states to visit on each on the remaining three days. They have one visit per day, and they have to change states every day, so either they visit all three states, or they visit one twice and one once.

   For parts (a) to (d) we also assume that the first state has 7 votes, the second state 8, and the third one 13.

   • a) Assume each of the three state has seen each candidate equally often so far. What will the candidates do?
   • b) Assume the small state has seen Ann once more than Beth so far, the medium state has seen Beth once more than Ann, and that the large state has seen both candidates equally often. What will both candidates do in the remaining three days?
   • c) Assume the small state has seen Ann twice more than Beth so far, the large state has seen Beth once more than Ann, and that the medium state has seen both candidates equally often. What will both candidates do in the remaining three days?
   • d) Assume the medium state has seen Ann once more than Beth so far, the large state has seen Beth once more than Ann, and that the small state has seen both candidates equally often. What will both candidates do in the remaining three days?
   • e) How will (a), (b), (c), and (d) change if the small states has 6 votes instead of 7?
   • f) How will (a), (b), (c) and (d) change if the small states has 5 votes instead of 7?

   The recommendation is to find first the possible moves for both players, then compute the Normal form, and then eliminate weakly dominated moves to get the IEWD matrix.
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