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