^{1}

^{2}

In this note we analyze a game of marital infidelity. The husband can either be faithful to or cheat on his wife. The wife can either monitor or not monitor her husband. We first determine the best response correspondences of the two players. Second, we explain why there is no pure-strategy Nash equilibrium in the game under study. Third, we show that there exists a unique mixed-strategy Nash equilibrium in the game. Finally, we demonstrate the nexus between our marital infidelity game and the prominent Matching Pennies game.

The now deceased American actor and comedian Rodney Dangerfield once said “I told my wife the truth. I told her I was seeing a psychiatrist. Then she told me the truth: that she was seeing a psychiatrist, two plumbers, and a bartender.” The humor notwithstanding, the available evidence discussed by Parker-Pope [

At least since the appearance of the monopolistic competition model in Dixit and Stiglitz [

In a thoughtful paper, Fair [

When monitoring infidelity is difficult, the work of Sohn [

Adamopoulou [

Our understanding of marital infidelity has definitely been enhanced by the studies discussed in the foregoing three paragraphs. Even so, two points are now worth emphasizing and these two points together explain why the research conducted in this note is significant. First, the existing studies are mainly empirical and not theoretical in nature. Second, even though the marital infidelity phenomenon clearly has strategic aspects to it, with the exception of Batabyal [^{1}. Our analysis differs from Batabyal’s [

Given the above described lacuna in the literature, in our note, we focus on a married couple and analyze a game of marital infidelity. The husband can either be faithful to or cheat on his wife. The wife can either monitor or not monitor her husband. Section 2.1 describes the static game model^{2} we utilize to conduct the analysis. Section 2.2 determines the best response correspondences of the two players. Section 2.3 explains why there is no pure-strategy Nash equilibrium in the game under study. Section 2.4 solves for the unique mixed-strategy Nash equilibrium in the game between the husband and the wife. Section 2.5 demonstrates the nexus between our marital infidelity game and the Matching Pennies game. Section 3 concludes and then offers two suggestions for extending the research described in this note.

Consider a man and a woman who are married and live together in the same house. This couple does not have any children and hence 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 may cheat on his wife^{3}. The husband (player 1) must choose whether to be faithful to his wife or to cheat on her. Therefore, the two pure strategies for the husband are denoted by F for faithful and C for cheat. The wife (player 2) must decide whether or not to monitor her husband’s activities. As such, her two pure strategies are denoted by M for monitor and D for do not monitor.

We now need to specify 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 will need to capture the oppositional nature of the game-theoretic interaction between the husband and the wife in the context of marital infidelity. Keeping this point in mind, we suppose that the game of interest can be described in matrix form in

Husband (Player 1) | Wife (Player 2) | ||
---|---|---|---|

Monitor | Don’t Monitor | ||

Faithful | 1, 1 | 1, 2 | |

Cheat | 0, 2 | 2, 1 |

With this background in place, our next task is to determine the best response correspondences of the two players.

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

Some thought tells us that the husband will remain faithful to his wife (play F ) and not cheat (play C ) as long as his payoff from remaining faithful exceeds his payoff from cheating. Mathematically, straightforward algebra indicates that this happens when

u 1 ( F , q ) > u 1 ( C , q ) ⇔ 1 > 2 ( 1 − q ) ⇔ q > 1 / 2 . (1)

Using the logic leading to (1), it is clear that the husband will prefer to cheat and not remain faithful if and only if q < 1 / 2 . Finally, when q = 1 / 2 the husband will be indifferent between remaining faithful to and cheating on his wife.

Moving on to the incentives confronting the wife, we see that she will prefer to monitor her husband’s activities (play M ) provided that her payoff from monitoring is greater than her payoff from not monitoring (playing D ). Once again, elementary algebra tells us that monitoring will occur when

u 2 ( M , p ) > u 2 ( D , p ) ⇔ p + 2 ( 1 − p ) > 2 p + ( 1 − p ) ⇔ p < 1 / 2 . (2)

From the result in (2) it follows that the wife will not monitor her husband’s activities when p > 1 / 2 and that she will be indifferent between monitoring and not monitoring when p = 1 / 2 .

Using the analysis in the preceding two paragraphs, we can express the best response correspondences of the husband and the wife. To this end, let R 1 * ( q ) and R 2 * ( p ) de- note the husband’s and the wife’s best response correspondences. We get

R 1 * ( q ) = { p = 0 if q < 1 / 2 p ∈ [ 0 , 1 ] if q = 1 / 2 p = 1 if q > 1 / 2 (3)

and

R 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 in this note is to point out that a pure-strategy Nash equilibrium does not exist in the game between the husband and the wife that we have been studying thus far.

Inspecting

To find the mixed-strategy Nash equilibrium, note from Equation (3) and our analysis thus far that the husband (player 1) will be willing to mix between his two pure strategies F and C if and only if u 1 ( F , q ) = u 1 ( C , q ) and this last condition holds only when q = 1 / 2 . Similarly, Equation (4) and some thought together tell us that the wife (player 2) will be willing to mix between her two pure strategies M and D if and only if u 2 ( M , p ) = u 2 ( D , p ) . This last condition holds only when p = 1 / 2 . Putting this information about the probabilities p and q together, we conclude that there exists a unique mixed-strategy Nash equilibrium ( p , q ) = ( 1 / 2 , 1 / 2 ) in the husband- wife game that we have been studying here. Specifically, in this game, the husband is faithful to his wife with probability p = 1 / 2 and his wife chooses to monitor his activities also with probability q = 1 / 2 4 . Our last task in this note is to demonstrate the nexus between our marital infidelity game and the well-known Matching Pennies game.

In the Matching Pennies game, players 1 and 2 each put a penny on a table simultaneously. If the outcome is two heads or two tails then player 1 gets to keep both pennies. Otherwise, player 2 keeps the two pennies. This game is well known and is routinely discussed in game theory textbooks such as Tadelis ( [

The point to note is that the incentives facing the husband and the wife in our marital infidelity game and the incentives facing the two players in the Matching Pennies game are very closely related. Specifically and consistent with the discussion in section 2.3, in both games, we have a situation in which one player wants to match the action of the other player but the other player wants to countermand this matching. This is also why in both these games, there is no pure-strategy Nash equilibrium and the only Nash equilibrium is in mixed strategies. This completes our game-theoretic analysis of marital infidelity.

In this note we provided one of the first game-theoretic studies of marital infidelity. Since our objective in this note has been to report the results of abstract theoretical research, the question of seeking “truth from facts” does not arise. Specifically, we analyzed a game model of cheating on one’s spouse. The husband could either be faithful to or cheat on his wife. The wife could either monitor or not monitor her husband. We first ascertained the best response correspondences of the two players. Second, we explained why there was no pure-strategy Nash equilibrium in the game under study. Third, we showed that there existed a unique mixed-strategy Nash equilibrium in the game. Finally, we pointed out the nexus between our marital infidelity game and the well-known Matching Pennies game.

The analysis in this note can be extended in a number of directions. Here are two suggestions for extending the research described here. First, it would be useful to introduce income heterogeneity between the husband and the wife into the model and to then study how this heterogeneity influences the likelihood of cheating either by the husband or by the wife. Second, it would also be interesting to introduce children into the model and then study marital infidelity when the husband and/or the wife care not only about their own welfare but also about the welfare of their children. Game-theo- retic studies of marital infidelity that incorporate these features of the problem into the analysis will provide additional insights into a phenomenon that has salient economic and psychological ramifications for a non-trivial proportion of society.

We thank an anonymous reviewer for helpful comments on a previous version of this paper. In addition, Batabyal acknowledges financial support from the Gosnell endowment at RIT and he thanks Cassandra Shellman for her assistance in formatting the paper. The usual disclaimer applies.

Batabyal, A.A. and Beladi, H. (2017) Cheating on Your Spouse: A Game-Theoretic Analysis. Theoretical Economics Letters, 7, 79-85. http://dx.doi.org/10.4236/tel.2017.71008