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In this paper, we study a game model of marital cheating. The husband is the cheater and the wife is faithful. The husband’s cheating is either open or surreptitious. The wife can either ignore the cheating or catch her husband in the act of cheating. We first express the game of interest in matrix form. Second, we determine the best response functions of the two players. Third, we show that there exists a unique mixed-strategy Nash equilibrium in the game. Finally, we demonstrate the nexus between our marital cheating game and the prominent Matching Pennies game.

In a humorous retort, the now departed American actor and comedian Rodney Dangerfield said “My marriage is on the rocks again, yeah, my wife broke up with her boyfriend.” The wit notwithstanding, several writers such as Ali [

We can think of the phenomenon of marital cheating as a love for one kind of variety in one’s life. Having said this, it is helpful to recall that economists have studied a love for variety and the impact that this love has on consumer welfare in considerable detail at least since the seminal work of Dixit and Stiglitz [

The reader should recognize that the extent of cheating in marriages in the United States is not inconsequential by any means. In addition to the findings discussed by Ali [

In a dated but nonetheless thoughtful paper, Fair [

Sohn [

Two empirical findings based on United States data in Adamopoulou [

The studies discussed in the preceding three paragraphs have certainly advanced our understanding of cheating in marriages. Even so, two points are now worth stressing. First, the existing studies are mainly empirical and not theoretical in nature. Second, even though the phenomenon of cheating in a marriage has obvious strategic aspects to it, with the exception of Batabyal [^{1}. In the course of our analysis, we shall point to the connection between our “catching the cheating spouse” game and the well-known Matching Pennies game.

Given the above delineated lacuna in the literature, in our paper, we analyze a game model of marital cheating. In this model, the wife is faithful and the husband is the cheater. This cheating on the part of the husband is either open or surreptitious. The wife can either disregard the cheating or ensnare her husband in the act of cheating. Section 2.1 first describes the static game―see Gibbons [ [

Consider a man and a woman who are married and live together in the same house. This couple does not have any children and therefore they are the object of each other’s love and attention. In what follows, we assume that the wife is faithful to her husband but that the husband is cheating on his wife^{2}. The husband (player 2) must choose whether to cheat on his wife openly or to do so surreptitiously. Note that if the husband cheats on his wife openly then it is clear that he really does not care about keeping his marriage intact. As such, the two pure strategies for the husband are denoted by O for cheating openly and S for cheating surreptitiously. The wife (player 1) must decide whether to ignore her husband’s cheating or to catch him in the act of cheating. As such, her two pure strategies are denoted by I for ignoring the cheating and C for catching the cheating.

We now need to indicate the payoffs to the husband and to the wife from the pursuit of their two possible pure strategies. The reader should note that these payoffs need to account for the oppositional nature of the strategic relationship in the context of marital cheating. Bearing this point in mind, we suppose that the underlying static game of interest can be delineated in matrix form as follows:

In

Husband (Player 2) | |||
---|---|---|---|

Wife (Player 1) | Cheat Openly (O) | Cheat Surreptitiously (S) | |

Catch Cheating (C) | 20, −10 | 0, 0 | |

Ignore Cheating (I) | 10, 10 | 10, 0 |

the matrix consists of a pair of numbers. The first number refers to the row player’s payoff and the second number denotes the column player’s payoff. With this background in place, our next task is to ascertain the best response functions of the two players.

Let u 1 ( ⋅ , ⋅ ) and u 2 ( ⋅ , ⋅ ) denote the payoff functions of the wife (player 1) and the husband (player 2). In addition, let p ≥ 0 denote the probability that the wife selects C and therefore ( 1 − p ) ≥ 0 denotes the probability that she selects I . Similarly, let q ≥ 0 denote the probability that the husband chooses O and hence ( 1 − q ) ≥ 0 denotes the probability that he chooses S .

The structure of the individual payoffs in the game depicted in

u 1 ( C , q ) > u 1 ( I , q ) ⇔ 20 q > 10 ⇔ q > 1 / 2 . (1)

Utilizing the logic leading to (1), it should be clear to the reader that the wife will prefer to ignore her husband’s cheating and not attempt to catch him as long as q < 1 / 2 . Finally, when the probability q = 1 / 2 , the wife is indifferent between attempting to catch her husband cheating and ignoring this cheating.

Let us now focus on the incentives confronting the husband. It is straightforward to confirm that instead of cheating on his wife surreptitiously (playing S ), the husband will prefer to cheat on her openly (play O ) if and only if his payoff from cheating openly exceeds his payoff from cheating surreptitiously. Mathematically, this means that we must have

u 2 ( p , O ) > u 2 ( p , S ) ⇔ − 10 p + 10 ( 1 − p ) > 0 ⇔ p < 1 / 2 . (2)

The inequalities in (2) tell us that the husband will prefer to cheat on his wife surreptitiously when the probability p > 1 / 2 and that he will be indifferent between cheating openly and surreptitiously when this same probability p = 1 / 2 .

Using the analysis in the previous two paragraphs, we can express the best response functions of the wife and the husband. To this end, let B 1 ∗ ( q ) and B 2 ∗ ( p ) denote the wife’s and the husband’s best response functions. Then, using our analysis thus far, we get

B 1 ∗ ( q ) = { p = 0 if q < 1 / 2 p ∈ [ 0 , 1 ] if q = 1 / 2 p = 1 if q > 1 / 2 } . (3)

And

B 2 ∗ ( p ) = { q = 1 if p < 1 / 2 q ∈ [ 0 , 1 ] if p = 1 / 2 q = 0 if p > 1 / 2 } (4)

Our next task is to solve for the unique mixed-strategy Nash equilibrium in the game between the cheating husband and the loyal wife.

As a prelude to solving for the mixed-strategy Nash equilibrium, we would first like to point out that there is no pure-strategy Nash equilibrium in the marital cheating game that we have been studying thus far. To see this clearly, let us revisit

Now, moving on to the unique mixed-strategy Nash equilibrium, observe from Equation (3) and our analysis thus far that the wife (player 1) will be willing to mix between her two pure strategies C and I if and only if u 1 ( C , q ) = Math_40# and this equality holds only when the probability q = 1 / 2 Similarly, equation (4) and some thought tell us that the husband (player 2) will be willing to mix between his two pure strategies O and S if and only if u 2 ( O , p ) = u 2 ( S , p ) . This last condition holds only when the probability p = 1 / 2 . Collecting this information about the two probabilities p and q , we deduce that the unique mixed-strategy Nash equilibrium of our game is where ( p , q ) = Math_49#. Put differently, in the game under study, the husband cheats on his wife openly with probability q = 1 / 2 and the wife catches him cheating openly with probability p = 1 / 2 . Our final task in this paper is to establish the relationship between our marital cheating game and the well-known Matching Pennies game.

In the Matching Pennies game, players 1 and 2 each place a penny on a table concurrently. If the resulting outcome is two heads or two tails then player 1 gets to keep both the pennies. If this is not the resulting outcome then player 2 keeps the two pennies. This game is well known and is standardly discussed in game theory textbooks such as Gibbons [ [

The reader should note that the incentives facing the husband and the wife in our marital cheating game and the incentives confronting the two players in the Matching Pennies game are very closely linked. In this regard, consistent with the discussion in section 2.3, note that in both games being compared, we have a scenario in which one player wants to match the action of the other player but the other player wants to get around this matching. As noted previously, this is also why a pure-strategy Nash equilibrium does not exist in both these games and the only Nash equilibrium is in mixed-strategies. This completes our game- theoretic perspective on catching a cheating spouse.

In this paper, we analyzed a game model of marital cheating. Our analysis did not focus on the phenomenon of divorce in a marital relationship and nor did it address the fact that the interaction between the husband and the wife in a marriage typically occurs repeatedly over time. Therefore, it would be useful to analyze how the possibility of divorce affects the cheating behavior of the husband or, more generally, either spouse in a marital relationship. Second, it would also be interesting to study how the incentives confronting the husband and the wife in the “cheating and catching” static game analyzed in this paper change when the underlying game is repeated over time. Game-theoretic studies of marital cheating that incorporate these features of the problem into the analysis will provide additional insights into a phenomenon that has significant economic and societal implications for a non-negligible proportion of society.

Batabyal thanks two anonymous reviewers for their helpful comments on a previous version of this paper and Cassandra Shellman for her help in formatting the final version of the paper. In addition, he acknowledges financial support from the Gosnell endowment at RIT. The usual disclaimer applies.

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