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Game Theory and Literature
Game-theoretical literary models employ a productive set of emerging perspectives on social interaction that supplement traditional exegetical methods, drawing on thorough analyses of literary structures and contexts in order to open
new avenues of inquiry. Scholars have recently explored such methods' in feminism and ethics, African-American studies, and a broad range of individual authors and genres. Studies of game dynamics in the Old Testament, Shakespeare, Puccini, Arthur Con
an-Doyle, and others have opened cultural and literary study to the insights of social science that multiple voices and perspectives have clearly been recognizing and implementing for millennia . As Peter Swirski notes in "The Role of Game Theory in Lite
rary Studies," the game-theoretical perspective addresses "the narrative richness and complexity of literary works, while at the same time acknowledging the fundamental nature of the conflicts they are modeled upon" (42).
The Prisoner's Dilemma
Imagine that Sir Gawain and Sir Kay have both been apprehended by Morgan le Fay for trespassing in the environs of the Green Chapel. By dint of her sorcery, they are magically transported to separate cells in a dungeon in her remote hideout in Avalon. Merlin is appointed their attorney, and informs them each separately that they have two possible courses of action, and that their fates each depend on the other's choice. Gawain may choose to "defect" on Sir Kay, implicating him in some trumped up charges-let's say stealing holy relics from the chapel-or he may "cooperate" with his accomplice and reveal nothing. Kay has the same options. Both knights also know the following: If Gawain defects on Kay, Gawain will be transported to a lush garden vale where he will be attended by one hundred of the realm's most beautiful maidens; and, as long as Kay cooperates, he will be drawn and quartered. The opposite scenario will take place if Kay defects and Gawain cooperates. If they both defect and rat each other out, they will both be branded with the Seal of Un-Chivalry and sent back to Arthur's court, alive but doomed to enduring humiliation. If they both cooperate, they will be released after a brief sentence of dragon-slaying in the nearby forest, which won't be so bad, because they will finally have a chance to do deeds of prowess.
Here's where it gets tricky. From both knights' points of view, the possible outcomes are ranked in the following order: 1) Vale of Maidens (we'll call this the Temptation to defect); 2) Dragon Slaying (Reward for mutual cooperation ); 3) Branded with Seal of Un-Chivalry (Punishment for mutual defection); and 4) Drawing and Quartering (Sucker's payoff). The bad news is that defection looks like the most logical alternative from either player's point of view-regardless of what the ot her does. If Gawain were to cooperate, Kay would surely reason that defection and transportation to the vale of maidens is the best choice. On the other hand, if Gawain defects, Kay would reason, he too must choose to defect, because even the Seal of Un -Chivalry is better than drawing and quartering. Kay thinks it over, and realizes that defection is the best choice for him regardless of what Gawain decides. The situation looks grim indeed for our knights, who will each be better off defecting, regardl ess of the other's choice.
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In a game theoretical context these punishments and rewards would be assigned values such as: T=5, R=3, P=1 and S=0 (see figure 1). From the point of view of either prisoner, it seems impossible to earn the mutual reward for simultaneous cooperation because no further consequences ensue. If they could see their situation from the perspective of the group, ,utual cooperation would be seen to raise not only their aggregate payoff but their chances of both coming out alive. Ho wever, since this game is a one-shot dilemma, neither has any particular incentive to cooperate: cheated opponents will not be able to retaliate later. Nevertheless, as we know from Arthurian literature, knights face similar challenges over and over agai n, with each challenge testing prowess and chivalry. What if there were a magic salve (like the one in Malory's Tale of Sir Gareth) that could be used to stick Gawain or Kay back together again? Or, what if they had to repeat similar situations indefinit ely? Would their choices change if they knew they might face similar situations in the future?
According to a study by Robert Axelrod, indefinite repetition is the key to establishing reciprocally cooperative relations between players of the prisoner's dilemma (12-13). In any definitely limited set of games, both parties will be tempted to defect on the last round, because they know they will never meet again after the last round, and that defection is the best strategy from the individual point of view. Therefore, each player will reason, because each knows that the other w ill defect on the last round, defection becomes the best alternative on the next to last round, and so on, down to the very first round (Axelrod 10, Dutta 214). However, if neither player knows when the rounds of prisoner's dilemma games will end, each w ill be better served by concentrating on maximizing the payoff of the current round. After all, anything could happen in the future, but a minimized penalty or a maximized reward in the present seems far more pressing. In fact, due to a sort of "inflati on," the weight with which players consider future payoffs should be discounted, for example, by a factor of one half the weight of the previous game. The weight of the first round (round 0) can be set at 1, and the weight of consideration for future pot ential rounds will be discounted by ½ to the power of the number of the round. Thus rounds are weighted sequentially, multiplying successive payoffs by 1, ½, ¼, 1/8, etc. to the last round (which will take place at some indefinite time in the future). I f the inflation adjustment is sufficiently high (ie future rounds are weighted as a high percentage of current payoff value), maximizing present payoff through mutual cooperation is the best plan, since its immediate value plus an indefinite run of discou nted cooperation rewards is greater than a temptation reward plus a discounted run of punishments (Axelrod 12-13). For example, if the discount of future rounds is set to .6, the sum of rewards for mutual cooperation exceeds that of an initial temptation plus subsequent mutual defection punishments by the fourth round . However, the incentive for the players to continue cooperating in any given round remains the fact that they are unsure of when the games will end.
Now imagine that our knights are faced with indefinite series of situations in which they are matched with opponents about whom they have very little information and cannot anticipate whether they will cooperate (behaving courteously , at their first meeting, for example) or defect (demanding combat). What would be the strategy that would best serve well-meaning knights in an uncertain world? According to the research of Axelrod (1981) and Nowak and Sigmund (1993 and 1998), generall y cooperative strategies will triumph in many contexts, although no strategy is infallible. The most widely known of these strategies is called 'tit-for-tat.' Axelrod explains that:
TIT FOR TAT is the policy of cooperating on the first move and then doing whatever the other player did on the previous move. This policy means that TIT FOR TAT will defect once after each defection of the other player. . . . The first question you are tempted to ask is, "What is the best strategy?" In other words, what strategy will yield a player the highest possible score? This is a good question, but . . . no best rule exists independently of the strategy being used by the o ther player. In this sense, the iterated prisoner's dilemma is completely different from a game like chess. . . . The interests of the players are not in total conflict. Both players can do well by getting the reward, R, for mutual cooperation, or both can do poorly by getting the punishment, P, for mutual defection. . . .
In fact, in the Prisoner's Dilemma, the strategy that works best depends directly on what strategy the other player is using and, in particular, on whether this strategy leaves room for the development of mutual cooperation. (13-15)
In Axelrod's tournaments, virtual players employing dozens of strategies were pitted against one another in repeated iterations of the prisoner's dilemma. In both tournaments, tit-for-tat emerged as the winner. Subsequent studies b y Nowak and Sigmund (1993, 1998), have discovered even more effective strategies that essentially replicate the cooperative tendencies of tit-for-tat, with occasional defection against opponents who always cooperate or speculative cooperation despite acci dental mutual defections. The significance of this research lies in the finding that "cooperative social behavior can emerge as the result of evolutionary processes involving individually selfish agents" (Casti 76). Ultimately these strategies are consi dered 1) Nice: they do not defect at first; 2) Forgiving: they will reciprocate if an opponent cooperates; 3) Responsive: moves are based on opponents' actions; 4) Provocable: they will retaliate in turn for defection; and 5) Transparent: a limited number of rules govern conduct (Méró 41-42). Nowack's and Sigmund's studies confirm the regularity with which analogous strategies emerge in evolutionary contexts where virtual players' strategies change over time.
It is worth repeating, however, that no pure strategy is completely effective in all contexts, and that even "nice" strategies like tit-for-tat may become trapped in cycles of mutual defection. What's more, in populations of players whose strategies evolve over time-increasing or decreasing their proportions based on the relative strength of their scores-strategies like tit-for-tat are susceptible to "invasion" by other "nice" strategies, such as that of cooperating on every move or "All C" (Casti 77). Tit-for-tat will never compete with All-C players, allowing them to proliferate. When a sufficient proportion of All-C players has emerged, strategies that are "mean," such as All-D players (who always defect) can gain a foothold, b ecause their interactions with All-C players enable them to win the benefits of multiple temptation payoffs. Nowack and Sigmund (1993) discovered a strategy of "win-stay, lose-shift" (dubbed 'Pavlov' for its simple responsiveness) that repeats its last m ove if rewarded with R or T and does the opposite if it earns P or S: "Thus Pavlov is fairly tolerant, and can correct mistakes . . . has no qualms about exploiting a sucker (after an accidental mistake [sic]), [and All-C players] cannot subvert a Pavlov population" (56-7).
Click here for strategy in Sir
Gawain and the Green Knight and Malory's
Arthuriad.
Acknowledgements and Credits
