Consider the “Odd Couple” game. Felix and Oscar share an apartment, the state of

Question

Consider the “Odd Couple” game. Felix and Oscar share an apartment, the state of
cleanliness of which is a public good. It takes 12 hours of work per week to make the
apartment spotlessly clean, 9 hours to be livable, and anything less leaves the
apartment in a state that would not stand inspection by the local rodent police. Felix
and Oscar each get a (gross) payoff of 2 from a livable apartment, but Felix is a
fusspot who assigns a payoff of 10 to a spotless apartment whereas Oscar gets a
payoff of only 5. A filthy apartment is worth -10 to Felix but only – 5 to Oscar. Each
person’s net payoff equals his respective gross payoffs minus his respective hours
worked cleaning. The matrix of net payoffs for the game is as follows.


Oscar
3 hours 6 hours 9 hours
Felix 3 hours -13, - 8 - 1, - 4 7, - 4
6 hours - 4, - 1 4, - 1 4, - 4
9 hours 1, 2 1, - 1 1, - 4


Part a. What does Iterated Elimination of Dominated Strategies (IEDS)
predict will be the outcome to the odd couple game if “dominated strategies” in the
definition of IEDS refers to strict domination.

Part b.What does Iterated Elimination of Dominated Strategies (IEDS)
predict will be the outcome to the odd couple game if “dominated strategies” in the
definition of IEDS refers to weak domination?

Part c. Identify the Nash equilibria of the game. Explain your reasoning.

Part d.What general conclusions about the appropriate form of
“dominated strategies” in determining the set of rationalizable strategies for players
and corresponding predicted outcomes in any given game arise from your answer to
the above? Explain.

Explanation / Answer

Strict dominance means that a certain choice is always worse than the rest of the alternatives no matter what action your opponent takes. Looking at the pay off table, we see that for Oscar, 9 hours of work always gives less of a payoff than 3 or 6 regardless of what Felix chooses, so we can eliminate that strategy. Having eliminated that strategy, we see that for Felix, only working 3 hours provides less of a pay off than 6 or 9 (since he knows Oscar won't choose to work 9 hours, we can disregard this outcome), so we can eliminate that strategy too leaving us the reduced game of Oscar 3 6 Felix 6 -4,-1 4,-1 9 1, 2 1,-1 Now we can see that there are no more strictly dominated strategies to eliminate. Since Oscar can do no worse than -1 with any strategy, but possibly 2 if he works only 3 hours, he will choose that. Felix being aware of this will then choose to work 9 hours so the pay offs will be Felix 1 Oscar 2 b. Eliminating the strictly dominated strategies and using the payoff table from above, we can eliminate weakly dominated strategies which are those for which is at least one action your opponent can take that gives you a worse pay off than other strategies, but for all other actions, the pay off is at least as good. In this instance, for Oscar, working 3 hours gives him the same payoff as working 6 if Felix chooses to work 6, but will give him more if Felix chooses to work 9. Therefore, working 6 hours is weakly dominated by the strategy of working 3, since he can do no worse than a pay off of -1 under either strategy, but possibly better by choosing 3. There are no weakly dominated strategies for Felix. The result is the same as above, Oscar working 3 and Felix choosing 6. c. Nash equilibria exist where no player can do better off by unilaterally changing their strategy. If we are at Oscar 3 Felix 6, the pay off is 1 and 2 respectively. If Oscar decides to change to working 6 he is worse off, so will not switch. If Felix changes to working 6, he is also made worse off. So 3 6 is a Nash equilibria. Additionally, Oscar 6 and Felix 6 can be a Nash equilibria, since if Oscar were to switch to working 3, he would be no better off (unless Felix were to switch as well, but since we are looking at unilateral action, this possibility is irrelevant), and thus have no incentive to do so. Also, if Felix were to switch to working 9, he would be made worse off. So a Nash equilibrium exists here too. d. Strictly dominated strategies can be excluded from the set of rational strategies for players, since choosing them makes the player worse off in all circumstances (and is thus and irrational choice). Weakly dominated strategies however, remain in the set of rational strategies, since these can be an equilibrium where neither player has an incentive to switch. In this instance only the equilibrium of Felix 9 Oscar 3 is a Pareto optimal equilibrium since moving to the other equilibrium makes Felix better off at the expense of Oscar, but these strategies cannot be eliminated. d.

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