Game Theory And Cheating

The alternative is not to cheat on the agreement. Cheating increases a firm’s profits if its rival does not respond. Figure 11.7 “To Cheat or Not to Cheat: Game Theory in Oligopoly” shows the payoff matrix facing the two firms at a particular time. As in the prisoners’ dilemma matrix, the four cells list the payoffs for the two firms.
In game theory, cheating and cooperation are rational choices. If I cheat you today, I won’t be able to cooperate with you tomorrow. The theory is intuitively correct for bilateral relationships, but obviously doesn’t explain why a thousand people might choose to cooperate.

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Game Theory And Cheating Games

The same psychology is at work as the “win at any cost mentality.” In this mode, any devious method to outsmart your opponent, even when you are playing against yourself, is all right. Game theory can also be applied to romantic relationships (in this case, other than marriage). People may not realize it, but probability is a large factor in relationships. When things start to crumble, there is the probability that you could leave the person and get over them.

Applying game theory to social norms

By Peter Schuler
News OfficeIn his new book, Law and Social Norms, Eric Posner, Professor in the Law School, explores how the law must take into account the informal but powerful, cooperative relationships among individuals. Posner, who is editor of the

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Journal of Legal Studies, teaches contract law and bankruptcy law among other subjects.

Eric Posner

What is the gap in legal thinking that you are trying to help fill?

“Law professors and legal academics often think of the law as merely the regulation of behavior: making sure that people keep their promises in contracts, that businesses run smoothly, that crime is deterred and so forth. They tend to view this in very simplified formulas. Their view is that people are atoms and do what they want to, and then there is the government to enact laws and prevent people from engaging in antisocial behavior. On the contrary, social order is, to a certain extent, self-maintaining and you don’t always need a government involved. I hope this book encourages legal scholarship to focus on the importance of social norms in the decisions of legislators, judges and others who enact and enforce the law.”

What is the connection between the law and what you call “social norms”?

“One way to look at it is that much of our ordinary, everyday behavior is cooperative, even though we don’t expect the law to get involved. Even where societies are very weak, order is still maintained. Sociologists, economists, philosophers and historians have thought about this, but in legal scholarship, people don’t discuss it very much. It’s in the realm of nonlegal, “social” regulation, so for some law professors, it just complicates matters unnecessarily.”

How do you suggest that legal scholars view social norms?

“If people don’t commit crimes, because the neighborhood is well-organized and people are watching each other’s doorsteps, then there’s not much crime, and you really don’t need the law. Therefore, the assumption by some legal scholars might be that you really don’t need to talk about it. My argument is that, in fact, law sometimes can undermine nonlegal forms of regulation and sometimes enhance it. So when you’re evaluating law, you have to have a theory in the back of your mind about how nonlegal regulation occurs. If you see something bad, like people breaking promises or committing crimes, and you propose a law that will prevent this from happening, that’s fine. At that point, some law professor would make a prediction about the effect of the law. The additional step, however, is that you have to worry whether the law will change nonlegal sanctions.”

Would you discuss marriage and the family, which you examine in detail in your book and use as an example of an essential social institution that reflects both legal and cooperative arrangements?

“The case of marriage and the family is interesting because families are considered in many ways to be paradigmatic, self-regulating institutions. The traditional view has been that government should stay out of families. Contracts between husbands and wives are not enforceable; the law will not step in. Until recently, if abuse occurred in the family or one member committed a tort against another, there was no legal redress. Wives and husbands and others in the family couldn’t sue one another. That’s the traditional approach. The intuitive argument is that the government doesn’t know the best way to regulate the family; the family is remote, complicated and intimate, and the government will only make things worse. (The family isn’t the only sovereign institution like this. For example, religious organizations in the United States sometimes operate outside the realm of government.) Yet government does regulate the family heavily, but only at its beginning and its end: in the marriage contract and in divorce. That particular significant regulation, where you need the government’s permission to get married, contrasts sharply with the government’s hands-off approach to commercial contracts. The law is set up so that usually anyone can contract with anyone else at any time. The government only steps in when a dispute arises to figure out who was right and who was wrong.”

Describe how a legislator might analyze a family issue using your approach.

“In the case of the family, the concern is that all sorts of bad things happen, like spousal abuse, and neglect and abuse of children. One view is that these are problems in which the government should intervene with police, injunctions and other tools of the law. But if the government intervenes too much, the family as an institution may fall apart because it needs autonomy to persist. Government over-regulation of the family will cause it to weaken and the institution may self-destruct.”

Could you explain how game theory informs your concept of social norms?

“Game theory provides a model for social organizations and cooperation between individuals. It assumes that people are rational actors. Game theory uses a theoretical construct that sets aside affection or altruism or a sense of solidarity as factors in cooperation. In the classic Prisoner’s Dilemma of game theory, people will rationally cheat each other and engage in opportunism unless something prevents them from doing so. In game theory, cheating and cooperation are rational choices. If I cheat you today, I won’t be able to cooperate with you tomorrow. The theory is intuitively correct for bilateral relationships, but obviously doesn’t explain why a thousand people might choose to cooperate. If it did, then we’d have lots of cooperation and very little government, because people would always fear that someone would see them acting badly (cheating) and tell others, so no one would rationally choose to cheat.

“How does the game-theory concept of discount rates fit into this?”

“In simple terms, people have low discount rates when they care about the future. People who are mature and patient will value future payoffs. People who cannot defer gratification want payoffs now: high discount rates. In my book, I call them “good types” and “bad types.” Another game-theory concept, “signaling,” refers to taking costly actions as a way of revealing private information. If you know what investments you must make to assure the cooperation of others, you will make those investments. My concept of social norms refers to those behaviors people take to signal that they are good types. Examples might include how you dress, owning a nice car and giving to charity. Cooperation is generally desirable, and signaling is part of that. Social norms refer to the patterns of behavior that emerge as people signal their time preferences and other kinds of private information.”

Explain how crime shows how signaling and social norms should be part of government decisions.

“The government historically used shame to deter crime. Criminals were publicly identified and humiliated, and even branded. But once marked with a brand as a criminal, an individual was shunned by society and had no choice but to join a criminal gang. That’s a case of government exploitation of social norms that has an unintended and negative result. The government is always faced with these choices of using laws or social norms. It can punish criminals to deter them or it can take advantage of social norms as signals, with unpredictable outcomes.

“So when the government sees a social problem, there are basically two strategies: traditional sanctions to punish bad behavior and reward good behavior, or enhancing the power of a group whose social norms might independently solve the problem. There’s a trade-off either way, but the choice is the first step in figuring out how to solve the problem.”

How to catch a cheating spouse using game theory

Unfortunately, marital cheating exists and occurs everywhere. Some spouses pretend not to know and ignore it, while others search for ways to catch their significant other in the act. According to the article, “between 30 and 60 percent of all married persons in the United States will engage in cheating at some point in their marriages.” Interestingly, the article states that men and women are equally likely to cheat on their spouse, and having an affair tends to have similar impacts on women and men.

In this marital cheating game, the husband is cheating on his wife, while the wife remains faithful. The husband (player 2) cheats openly or in secret; thus, his two pure strategies are O (cheat openly) and S (cheat in secret). Meanwhile, the wife (player 1) either ignores the cheating or catches him in the act, and her two pure strategies are I (ignore the infidelity) and C (catch him cheating).

It is important to note, as the article states, that the numbers in the payoff matrix above account for the “oppositional nature of the strategic relationship in the context of marital cheating.” After looking at the payoff matrix, I can see that there is no pure-strategy Nash Equilibrium, which in lecture was defined as a pair of strategies in which each player’s strategy is a best response to the other player’s strategy. There is no pure-strategy Nash Equilibrium because if the wife chooses to catch her husband cheating (C), then the husband would choose to cheat in secret (S). However, if the husband chooses to cheat in secret (S), then the wife would choose to ignore the cheating (I). These two are not best responses. Likewise, if the wife were to ignore the cheating (I), then the husband would choose to cheat openly (O). However, if the husband were to cheat openly (O), then the wife would choose to catch him cheating (C). Again, we see that these are not best responses to each other.

However, we can find a mixed-strategy equilibrium. Using the principle of indifference in the textbook, I let p be the probability that the wife catches him cheating (C), and I let q be the probability that the husband openly cheats on his wife (O). Thus, letting 1-p be the probability that the wife ignores the cheating (I), and 1-q being the probability that the husband secretly cheats (S). In a mixed strategy equilibrium, in order to make the player indifferent between the two strategies, it must be the case that the payoff from one strategy (let’s say catch cheating C) is equal to the payoff from the other strategy (let’s say cheat openly O). First, the husband chooses a probability of q for cheating openly O. Then, the expected payoff to the wife for strategy catch cheating C is (20)(q) + (0)(1-q), which equals 20q, while the expected payoff to the wife for ignore cheating is (10)(q) + (10)(1-q), which equals 10. To make the wife indifferent between the two strategies, we need to set 20q=10, which leaves us with q =1/2. Next, the wife chooses a probability of p for catching her husband cheat C. Then, the expected payoff to the husband for cheating openly is (-10)(p) + (10)(1-p), which equals to -20p + 10, while the expected payoff to the husband for secretly cheating is (0)(p) + (0)(1-p), which equals 0. To make the husband indifferent between the two strategies, we need to set -20p + 10 = 0, which leaves us with p = 1⁄2. Thus, the only possible probability values that can appear in a mixed-strategy equilibrium are p = 1⁄2 for the wife and q = 1/2 for the husband, and this in fact forms an equilibrium.

The fact that the probability for the wife to catch her husband cheating C is ½ and the probability for the husband to cheat openly is also ½ confirms the claim in the textbook that “we reach the natural conclusion that in any Nash Equilibrium, both players must be using probabilities that are strictly between 0 and 1.” As mentioned in the textbook, the reason for this is because both the wife and the husband want their behavior to be unpredictable to the other, so that their behavior cannot be taken advantage of. In this way, neither player’s behavior can be exploited by a pure strategy, and the two choices of probabilities (½ and ½) are best responses to each other. Thus, the claim that was mentioned in lecture by Professor Easley and mentioned in the book, “at least one mixed Nash Equilibrium must exist,” is indeed true.

This analysis reminds me of the attack-defense game mentioned in the textbook. In these attack-defense games, one player behaves as the attacker, while the other behaves as the defender. The attacker is able to use one of two strategies (A or B), while the defender’s two strategies are to defend against A or defend against B. If the defender defends against the attack the attacker is using, then the defender gets the higher payoff. However, if the defender defends against the wrong attack, then the attacker gets the higher payoff.

The marital cheating game is similar to the attack-defense game, where the husband is the attacker due to his infidelity, and the wife is the defender who remains faithful. Similar to the attacker, the husband is only able to use one of two strategies: cheat openly (O) or secretly cheat (S). Furthermore, if the wife catches her husband cheating while he cheats openly, she gets the higher payoff just as in the attack-defense game when the defender defends against the attack the attacker is using and gets the higher payoff.

Thus, it is evident that game theory and best responses play a crucial role in our everyday lives. A payoff matrix can always be developed by at least two individuals whose behavior depends not only on his or her behavior, but the behavior of the other individual as well. Every individual will try to maximize his or her payoff. The applications of game theory can represent cheating in different types of scenarios, including marital cheating. For instance, a game theory and payoff matrix can be developed for cheating on an exam, claiming unemployment benefits when one is not actually looking for work, making untrue statements on one’s tax returns, etc. Thus, evidently, game theory is widely applicable to our everyday lives.

Batabyal, A.A. (2017) A Game-Theoretic Approach to Catching a Cheating Spouse. Theoretical Economics Letters, 7, 464-470. https://doi.org/10.4236/tel.2017.73035

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