8 May 2013
Keeping the kitchen clean while living in a house with ten people is a frustrating day-to-day challenge. Ideally, everyone in the house cleans up each mess they make immediately after they make it. Instead, housemates often leave their dirty dishes in the sink or on the counter for hours waiting to wash them later. Even more frustrating, housemates may wait for someone else to get fed up with the pile of dirty dishes and proceed to wash them. The kitchen constantly accumulates dirty dishes. Irresponsible housemates continue to shirk on their responsibility to clean up their messes because their actions are unobservable, or they believe another housemate will need a clean kitchen and be forced to clean it. The tolerance level of the kitchen’s cleanliness differs between all ten of the housemates. Because of the different preferences, some are more inclined to wash the dirty dishes whenever, but others prefer to wait until there are literally no more clean dishes left leaving them no choice but to wash a dish. In creating a game theoretical model to try and better understand the quandary in our kitchen, we will use repeated prisoners’ dilemma games, collective action analysis, and finally, a two species evolutionary game.
We begin by applying a Repeated Prisoners’ Dilemma I (PDI) model to our situation. The rules for a PDI game are simple; the two players can either cooperate (wash) or defect (not wash). The players can fully observe each other’s move to wash their dish or to not wash. They will move in sequential order where player one moves first and the second player observes the move (see fig. 1). Then the second player carries out a move. The strategy called tit-for-tat (t-f-t), where player one cooperates on the first play and then does what the other player did the previous period, will be used to model the behavior of our players. The probability of continuing the game (p) is less than one, and so we have a finite game of unknown length. In our model, (p) is set at .9 showing that there is a ninety percent chance of continuation. We believe there is a ten percent chance that one player will not be using the kitchen to cook on the following day, which would end the current game. Using the t-f-t strategy will help us to understand when a player’s payoffs make sense to cooperate, defect once, or defect forever on dish duty given the discount rate (r) and the probability of continuation (p).
There are four possible payoff combinations in our PDI game where player one comes first in the payoff brackets (player 1, player 2) (refer to figure 2). The strategy where each player washes their dishes produces the social optimum payoff of twelve where payoffs are (6, 6) foffor the players. The social optimum is at (wash, wash) because it produces the highest possible utility of twelve. The next strategy is at (wash, no wash) which produces a payoff combination of (1, 10). The opposite strategy is at (no wash, wash) where the payoff combination is (10, 1). The final strategy is the Nash Equilibrium where both players choose (no wash, no wash). The payoffs for this strategy are three for each player (3, 3). Also, the dominant strategy is for both to not wash regardless of the other’s actions. The social optimum strategy received a payoff of (6, 6) because both players are doing their dishes and are happy that both are cooperating. The next strategy where the payoffs are (1, 10) or (10, 1) are so lopsided because one person is doing all the washing while the other reaps the benefits. The washer has a positive payoff here because they are glad that the dishes at least got done. Notice that the combination of their payoffs is only eleven, which is short of the social optimum. The last numerical payoff of (3, 3) is justified because both players got away with not doing their dishes immediately for whatever reason. Payoffs...
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