please read the following pages

4. Application to Anthropology: Jamaican Fishing One important school of anthropological thought, known as functionalism, holds that customs, institutions or behavior patterns in a society can be interpreted as functional responses to problems which the society faces. One method, then, of understanding the organization of societies would be to identify problems and stresses, see what kinds of behavior would provide good solutions, and compare a society's behavior patterns to those solutions. For example, incest taboos can be interpreted as societal solutions to genetic problems caused by inbreeding. In the 1950's some pioneering anthropologists began to use game-theoretic ideas in the service of functionalism. For example, Moore [1957] proposed that one could interpret divination as a societal mechanism for implementing mixed strategy solutions to a game. Recall that in a game with a mixed strategy solution, it is crucial that strategies be selected randomly, with certain probabilities. There must be no pattern of strategy choices, even an unconscious one, which could be noticed and exploited by an opponent. We suggested making choices by using a chance mechanism, but a society might feel uncomfortable making important decisions this way. A way to obtain the same goal would be to have a shaman read the favorable course of action from a randomly generated pattern of caribou bones. Interpretive rules might evolve to select different alternatives with optimal probabilities. The first quantitative application of two-person game theory to an anthropo- logical problem was Davenport [1960], a classic and still controversial paper on Jamaican fishing. Davenport studied a village of two hundred people on the south shore of Jamaica, whose inhabitants make their living by fishing. The fishing grounds extend outward from shore about 22 miles. \"Twenty-six fishing crews in sailing, dugout canoes fish this area by setting fish pots, which are drawn and reset, weather and sea permitting, on three regular fishing days each week... The fishing grounds are divided into inside and outside banks. The inside banks lie from 5 to 15 miles offshore, while the outside banks all lie beyond ... Because of special underwater contours and the location of one prominent headland, very strong currents set across the outside banks at frequent intervals in both easterly and westerly directions. These currents . . . are not related in any apparent way to weather and sea conditions of the local region. The inside banks are almost fully protected from the currents.\" [Davenport, 1960] The captains of the canoes might conceivably adopt three different fishing strate- gies: 24 TWO-PERSON ZERO-SUM GAMES Inside: put all pots on the inside banks. Outside: put all pots on the outside banks. In-Out: put some pots on the inside banks, some pots on the outside. These different fishing strategies have a number of different advantages and disadvantages. Here are some of them: e Since travel times are longer, crews following the Outside or In-out strategies can set fewer pots. e When the current is running, it is harmful to outside pots in a number of ways. The bamboo floats marking the location of the pots are dragged underwater and the fishermen cannot find them; the pots are moved around on the bottom and may be smashed; fish in the pots may be killed by changes in temperature and other conditions induced by the current. e The outside banks produce higher quality fish, both in varieties and in size. In fact, if many outside fish are available, they may drive the inside fish off the market. e The Outside or In-out strategies require sturdier canoes. Inside fishermen often buy their canoes used from fishermen who go outside. Since Outside canoes are newer and sturdier, their captains dominate the sport of canoe racing, which is prestigious and offers large purses. Davenport gathered data to estimate the payoff for following each of the three possible strategies when the current is running, and when the current is not running: Current Run Not run Inside 17.3 11.5 Fishermen Outside | 4.4 20.6 In-Out 52 17.0 Game 4.1 The payoffs are average profits in English pounds per fishing month to the captain of the fishing canoe. Davenport notes that these estimates were made before he had any knowledge of game theory or plans to do a game-theoretic analysis, so they should be free of unconscious bias. However, knowing game theory, we can treat this as a 3 X 2 game, solve for the optimal fishing strategy and compare it to the actual fishing pattern of the villagers. The game does not have a saddle point and there is no dominance.' The diagram is shown in Figure 4.1. The lowest 'If you read Davenport's paper, you will notice that he had an incorrect understanding of the idea of dominance. 4. JAMAICAN FISHING 25 Run Not run Current Figure 4.1 Payoff diagram for Game 4.1 point on the upper envelope involves the Inside and In-Out strategies. Solving that 2 X 2 game we get an optimal strategy of 67% Inside, 33% In-Out for the fishermen; an optimal strategy of 31% Run, 69% Not run for the current; and a value of 13.3. The comparison of this game-theoretic solution with actual behavior is quite striking. First, no captains followed the Outside strategy, which the villagers characterized as entirely too risky. Second, in the period Davenport observed, 69% of the captains followed the Inside strategy and 31% followed the In-Out strategy. Third, even the current seemed to be following its optimal strategy fairly closely: estimates over a two year period were that it ran 25% of the time. The conclusion is that this society has adapted well to its natural and economic environment. Davenport's analysis went unchallenged for several years and was widely cited in the anthropological literature. However, Kozelka [1969] and Read and Read [1970] independently pointed out that there is a serious flaw in the analysis. The fishermen's opponent in this game is a natural phenomenon, the current. It is not a reasoning entity, and its behavior is not affected by what the fishermen do. In particular, it would not adjust its behavior to take advantage of non-optimal play by the fishermen. The fishermen's correct behavior in this context would be to use the Expected Value Principle. They should observe the mixed strategy 25% Run, 75% Not run being followed by the current, calculate the expected values of their various strategies, and follow the strategy with the highest expected value. These expected values are Inside: 29X 1734 5% 115=1298 Outside: .25 X (4.4) + .75 X 20.6 = 1435 In-Out: 25 X352+ .75 X 17.0 = 14.05. 26 TWO-PERSON ZERO-SUM GAMES All of the fishermen should fish outside! Perhaps this society is not so well- adapted after all. The key to the rebuttal of this criticism, as noted in Bagnato [1974], is the villagers' explanation of why they do not use the Outside strategy: it is too risky. The current is notoriously unpredictable. Even if on average it may run 25% of the time, in the short run of a year it might run considerably more or less. SuppoOSEONEEATINMNBSYONMERIME. The expected payoffs would be Inside: 35173+ .65%X 115 =0 Outside: .35 X (4.4)+ 65 X206 =11.85 In-Out: 35X 52+ .65X17.0=12.87. Suppose the village needs a certain minimum payoff, say 13 pounds per canoe per month, over the course of a year to avoid hardship or economic ruin. If the fishermen followed the Outside strategy the village could be ruined by a long run of bad behavior by the current. The advantage of the game theoretic minimax solution is that it guarantees an average income of at least 13.3 pounds per canoe per month regardless of what the current does. In this and similar situations it is this security property of the minimax solution which may make it desirable even when the opponent is not a reasoning entity. We will discuss this kind of \"game against nature\" at greater length in Chapter 10. Exercises for Chapter 4 1. By the Expected Value Principle, Outside is the best strategy if the current is running 25% of the time, while Inside is best if the current is running 35% of the time. Find a current percentage for which In-Out would be the best fishing strategy. 2. Suppose the payoff for In-Out with no current were 15.0 instead of 17.0. What would be the effect on the game-theoretic solution? On the value of the game? 4.1) If the current runs 30% of the time, In-Out has the highest expected value (13.46). To find such a percentage, look at Figure 4.1 and see where the In-Out line is uppermost. 4.2) The solution is now to fish Inside 81% of the time, Outside 19%. The In-Out strategy is inactive. The value of the game falls to 13.2, a very minor drop. If you haven't done so already, you should look at what effect the change had on the payoff diagram for the game