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Download PDF 1249 KB 11.22.17 WorldQuant Perspectives Shall We Play a Game? The study of game theory can provide valuable insights into how people make decisions in competitive environments, including everything from playing tic-tac-toe or chess to buying and selling stocks. WorldQuant, LLC 1700 East Putnam Ave. Third Floor Old Greenwich, CT 06870 www.weareworldquant.com WorldQuant Shall We Play a Game? Perspectives 11.22.17 JOHN VON NEUMANN WAS A MAN WITH MANY INTERESTS, FROM theory of corporate takeover bids and the negotiations of Britain formulating the mathematics of quantum mechanics to and the European Union over the application of Article 50. One of developing the modern computer. Luckily for the birth of modern the most spectacular applications was in the distribution of large game theory, he also had a passion for poker, an interest that portions of the electromagnetic spectrum to commercial users culminated in the monumental work Theory of Games and in an auction organized by the U.S. government in 1994. Experts Economic Behavior, written in collaboration with economist in game theory were able to maximize both the government Oskar Morgenstern. Their focus was on cooperative games — revenue — amounting in the end to a staggering $10 billion — that is, games in which players can form coalitions or, in the and the efficient allocation of resources to the frequency buyers. words of Fields medalist John Milnor, “sit around a smoke-filled (A similar auction carried out by the New Zealand government in room and negotiate with each other.” Game theory was a radical 1990 without the help of game theory experts ended up being a departure from the standard view of economics, the so-called total fiasco.) Robinson Crusoe economy, in which consumers’ well-being is In this article we will outline the very basics of noncooperative not affected by their social interactions: Like Crusoe, alone on game theory with a view toward financial applications, starting a deserted island where he could interact only with nature, the from the ubiquitous prisoner’s dilemma and concluding with players in this idealized economy interact only with prices. But more-realistic games that describe important aspects of modern as von Neumann observed, “Real life consists of bluffing, of little financial markets and model real investors’ behavior. tactics of deception, of asking yourself what is the other man going to think I mean to do, and that is what games are about in Crime and Punishment my theory.” Alice and Bob have committed a serious crime: They stole $10 Von Neumann’s pioneering work had applications to warfare (he from their mom’s wallet, which she keeps in her nightstand. But was one source of inspiration for the iconic wheelchair-bound being amateur thieves, they were not too careful in orchestrating scientist in Stanley Kubrick’s Dr. Strangelove) but found limited their misdemeanor, and their mom, Carol, caught them applications in real world and economic theory. The next greatest trespassing in her bedroom. She immediately realized something revolution in game theory came from John Nash’s analysis of was out of order and checked her wallet, discovering with great noncooperative games, in which the emphasis is on individual consternation that money was missing. As a precautionary behavior. In 1994, Nash was awarded a Nobel Prize in economics measure, she decided to confine Alice and Bob to separate rooms. for the far-reaching applications of his groundbreaking work. Carol has enough evidence to convict both of her children for the Game theory has emerged as a very powerful and versatile lesser crime of trespassing, for which they would be grounded for technique, capable of modeling situations as diverse as the one day. But she lacks evidence for the principal crime, the theft. Being an ingenious woman, Carol offers Alice and Bob a deal. If they both confess to the theft, they will benefit from a reduced sentence and each will be grounded for five days. If neither The greatest revolution confesses, they will be sentenced only for the lesser crime and grounded for only one day. Finally, if Alice confesses while her in game theory came from accomplice remains silent, she will not face any charge and Bob John Nash’s analysis of will be grounded for 20 days. The same scenario applies if Bob noncooperative games, in confesses and Alice remains silent. The siblings must reach their which the emphasis is on decision independently, without communicating with each other. individual behavior. We can summarize Alice and Bob’s available strategies with the following payoff matrix: Copyright © 2017 WorldQuant, LLC WorldQuant Perspectives November 2017 2 WorldQuant Shall We Play a Game? Perspectives 11.22.17 Bob symmetrical, Bob will also choose to defect. The scenario in which both players defect is the sought-after Nash equilibrium for the silent confess prisoner’s dilemma. silent (-1,-1) (-20,0) Although neither player has an incentive to unilaterally change strategy, the Nash equilibrium does not represent the best Alice possible outcome: Alice and Bob would have been better off by confess (0,-20) (-5,-5) cooperating and remaining silent. But this scenario is not the equilibrium we just found. There is no “right” solution to this little game, hence the dilemma. Figure 1 The prisoner’s dilemma is the most paradigmatic example of a In the matrix the negative numbers represent the number of days non-zero-sum game. In this kind of game, complementary and of punishment. For example, the values in payoff (−5,−5) are the conflicting interests can be present simultaneously. In zero-sum days Alice and Bob would remain grounded if both confessed. games — like tic-tac-toe, chess or “global thermonuclear war” (played by the computer in the movie WarGames) — players From the siblings’ point of view, remaining silent and confessing are purely antagonistic. In these games “wealth” is transferred can be seen as forms of cooperation and defection. To highlight from loser to winner. In the financial world the futures, this interpretation, we relabel the payoff matrix as follows: options and currency markets are all zero-sum games. By contrast, the stock market is a non-zero-sum game because performance is inextricably linked to external factors, such as Bob the overall economic outlook. All investors could profit in a bull market, for example. cooperate defect Life is riddled with examples of the prisoner’s dilemma, from cooperate (-1,-1) (-20,0) countries negotiating on actions to limit global climate change to birds trying to remove ticks from each other’s feathers Alice (cooperation/defection corresponding to a bird agreeing/ defect (0,-20) (-5,-5) refusing to pull off its companion’s ticks). There is a simple and very practical financial application: competition in oligopolistic Figure 2 markets, where optimal quantity and price always depend on choices made by a small number of companies. Let us consider Given the assumption of rationality — and the adage that there is no honor among thieves — what strategies will Alice and Bob choose? To analyze their strategic behaviors, let us introduce the idea of the Nash equilibrium. We are in a Nash equilibrium if Life is riddled with Alice’s choice is optimal for Alice given Bob’s choice and at the same time Bob’s choice is optimal for Bob given Alice’s choice. In examples of the prisoner’s other words, neither player has an incentive to deviate unilaterally dilemma, from countries by playing a different strategy, given the strategy chosen by his or negotiating on actions her opponent. to limit global climate We can find the Nash equilibrium by first considering Alice’s point change to birds trying of view: If her brother defects, she can be punished with either 20 days (if she cooperates) or five days (if she defects). Given these to remove ticks from each outcomes, we can easily guess Alice’s choice: She will decide to other’s feathers. defect by confessing to her mom! Because the game is completely Copyright © 2017 WorldQuant, LLC WorldQuant Perspectives November 2017 3 WorldQuant Shall We Play a Game? Perspectives 11.22.17 the case of rivals Coca-Cola Co. and PepsiCo. It would be in the clear partiality for romantic comedies. Despite their personal interest of both cola makers to cooperate and keep the prices of preferences, they would rather watch a movie together than sit their carbonated beverages artificially high. But if, out of the blue, alone among strangers. Unfortunately, Alice’s cell phone battery Coca-Cola decided to reduce its price — that is, defect — PepsiCo is dead and they cannot communicate with each other. What should they do? would be forced to follow to protect its market share. We can represent the available options with the following payoff matrix, We can represent the situation with the following payoff matrix: in which the entries represent the increase in the companies’ profits per year (in arbitrary units). It is easy to see that we are again in a prisoner’s dilemma, as both companies have an Bob incentive to defect. action romantic PepsiCo action (4, 1) (0, 0) cooperate defect Alice romantic (0, 0) (1, 4) cooperate (5, 5) (0, 7) Figure 4 defect (7, 0) (2, 2) Coca-Cola We can think of the entries in the matrix as measures of Alice’s and Bob’s levels of happiness. For example, if they end up watching the action movie, the payoff can be interpreted as Figure 3 Alice being four times happier than her brother. The siblings’ unhappiness if they end up watching different movies is quantified In the prisoner’s dilemma we found a single Nash equilibrium.
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