A Theory of Marriage Vows


 Roger McCain's Game Theory: A Nontechnical Introduction to the Analysis of Strategy (Revised Edition) is available in Asia now and will be available in USA, UK, Europe and the rest of the World in October. Should you require the title for fall term, special arrangements can be made so please do contact Ms Hooi-Yean Lee (mkt@wspc.com). If you are considering to use this title after fall and would like to receive an inspection copy, please send your request via World Scientific Publishing Company's website. You may also visit http://www.worldscibooks.com/economics/7517.html for sample chapters.

Meanwhile, at a more advanced level, give some thought to McCain's Game Theory and Public Policy, Elgar, 2010. Part 1, "A Historical and Critical Survey," assesses the literature of game theory particularly from the point of view of the pragmatic purposes of public policy, while Part 2 presents some proposals and research toward a game-theoretic foundation for political economy. 
 This example is an attempt to use game theory to "explain" marriage vows. But first (given the nature of the topic) it might be a good idea to say something about "explanation" using game theory.

One possible objection is that marriage is a very emotional and even spiritual topic, and game theory doesn't say anything about emotions and spirit. Instead game theory is about payoffs and strategies and rationality. That's true, but it may be that the specific phenomenon -- the taking of vows that (in some societies, at least) restrict freedom of choice -- may have more to do with payoffs and strategies than with anything else, and may be rational. In that case, a game-theoretic model may capture the aspects that are most relevant to the institution of marriage vows. Second, game-theoretic explanations are never conclusive. The most we can say is that we have a game-theoretic model, with payoffs and strategies like this, that would lead rational players to choose the strategies that, in the actual world, they seem to choose. It remains possible that their real reasons are different and deeper, or irrational and emotional. That's no less true of investment than of marriage. Indeed, from some points of view, their "real reasons" have to be deeper and more complex -- no picture of the world is ever "complete." The best we can hope for is a picture that fits fairly well and contains some insight. I think game theory can "explain" marriage vows in this sense.

In some sequential games, freedom of choice can be a problem. We have seen this in the previous examples. These are games that give one or more players possibilities for "opportunism." That is, some players are able to make their decisions in late stages of the game in ways that exploit the decisions made by others in early stages. But those who make the decisions in the early stages will then avoid decisions that make them vulnerable to opportunism, with results that can be inferior all around. In these circumstances, the potential opportunist might welcome some sort of restraint that would make it impossible for him to act opportunistically at the later stage. Jon Elster made the legend of "Ulysses and the Sirens" a symbol for this. Recall, in the legend, Ulysses wanted to hear the sirens sing; but he knew that a person who would hear them would destroy himself trying to go to the sirens. Thus, Ulysses decided at the first stage of the game to have himself bound to the mast, so that, at the second stage, he would not have the freedom to choose self-destruction. Sequential games are a bit different from that, in that they involve interactions of two or more people, but the games of sequential commitment can give players reason to act as Ulysses did -- that is, to rationally choose at the first step in a way that would limit their freedom of choice at the second step. That is our strategy in attempting to "explain" marriage vows.

Here is the "game." At the first stage, two people get together. They can either stay together for one period or two. If they take a vow, they are committed to stay together for both periods. During the first period, each person can choose whether or not to "invest in the relationship." "Investing in the relationship" means making a special effort in the first period that will only yield the investor benefits in the second period, and will yield benefits in the second period only if the couple stay together. At the end of the first period, if there has been no vow, each partner decides whether to remain together for the second period or separate. If either prefers to separate, then separation occurs; but if both choose to remain together, they remain together for the second period. Payoffs in the second period depend on whether the couple separate, and, if they stay together, on who invested in the first period.

The payoffs are determined as follows: First, in the first stage, the payoff to one partner is 40, minus 30 if that partner "invests in the relationship," plus 20 if the other partner "invests in the relationship." Thus, investment in the relationship is a loss in the first period -- that's what makes it "investment." In the second period, if they separate, both partners get additional payoffs of 30. Thus, each partner can assure himself or herself of 70 by not investing and then separating. However, if they stay together, each partner gets an additional payoff of 20 plus (if only the other partner invested) 25 or (if both partners invested) 60.

Notice that the total return to investment to the couple over both periods is disproportionately greater if both persons invest -- that is, it is 2*20-2*30 in the first period plus 20+2*60 = 80 if both invest, but is 20+25=45 for one partner and 20 for the other partner if only one invests. The difference 80-65=15 reflects the assumption that the investments are complementary -- that each partner's investment reinforces and increases the "productivity" of the other person's investment.

These ground rules lead to the payoffs in Figure 19, in which "his" decisions are indicated by nodes labeled H for he, "her" decisions are denoted S for she, "her" payoffs are to the left in each pair and "his" are to the right. This complicated game of incomplete information has just four proper subgames, and all of them are basic. We will take the basic subgames in order from the top.

Exercise: Circle the four basic subgames of the game in Figure 19.

Since the decision to invest (or not) precedes the decision to separate (or not) we have to work backward to solve this game. Suppose that there are no vows and both partners invest. Then we have the first basic subgame as shown in Figure 20. This is a game of incomplete information and may be better analyzed by looking at it in the normal form, which is shown in Table 2. Clearly, in this subgame, to remain together is a dominant strategy for both partners, so we can identify 110, 110 as the payoffs that will in fact occur in case both partners invest.

Figure 19. Relationship

Figure 20. Basic Proper Subgame 1

Table 2. The Above Subgame in Normal Form

He

stay

go

She

stay

110, 110

60, 60

go

60, 60

60, 60

Now take the other symmetrical case and suppose that neither partner invests. We then have the subgame shown in Figure 21 and Table 3. Here, again, we have a clear dominant strategy, and it is to separate. The payoffs of symmetrical non-investment are thus 70,70.

Figure 21. Basic Proper Subgame 4

Table 3. Basic Proper Subgame 4 in Normal Form

He

stay

go

She

stay

60, 60

70, 70

go

70, 70

70, 70

Now suppose that only one partner invests, and (purely for illustrative purposes!) we consider the case in which "he" invests and "she" does not. We then have the subgame shown in Figure 22 and Table 4. Here again, separation is a dominant strategy, so the payoffs for the subgame where "she" invests and "he" does not are 115,40. A symmetrical analysis will give us payoffs of 40, 115 when "she" invests and "he" does not.

Putting these subgame outcomes together, we have reduced the more complex game to the one in Figure 23 and Table 5. This game resembles the Prisoners' Dilemma, in that non-investment is a dominant strategy, but when both players play their dominant strategies, both are worse off than they would be if both played the non-dominant strategy. Anyway, we identify 70, 70 as the subgame perfect equilibrium payoffs in the absence of marriage vows.

Figure 22. Basic Proper Subgame 2

Table 4. Basic Proper Subgame 2 in Normal Form

He

stay

go

She

stay

105,30

115,40

go

115,40

115,40

Figure 23. The Reduced Relationship Game

Table 5. The Reduced Relationship Game

 

He

invest

Don't

She

invest

110,110

40,115

Don't

115,40

70,70

But now suppose that, back at the beginning of things, the pair have the option to take, or not to take, a vow to stay together regardless. If they take the vow, only the "stay together" payoffs would remain as possibilities. If they do not take the vow, we know that there will be a separation and no investment, so we need consider only that possibility. In effect, there is now an earlier stage in the game. Using what we have already figured out and "reasoning backward," a partly reduced game will look like Figure 24.

Figure 24. A Partly Reduced Game with a Vow

We have already solved the lower proper subgame of this game. The upper proper subgame is simpler, since there are no go-or-stay games to be solved – having taken the vow, neither has the choice of going – but the payoffs here are taken from the stay-stay strategies in Figure 19. Once again, the upper proper subgame is a game of incomplete information, and it is shown in normal form in Table 6. This is a game with two Nash equilibria – where both sweethearts choose the same strategies, invest or do not invest – but since the "invest, invest" strategy is better for both, it is a Schelling point. If we identify the Schelling point at "invest, invest" as the likeliest outcome of this game, we have the fully reduced game shown in Figure 25.

Table 6. The Subgame with the Vow

He

invest

Don't

She

invest

110,110

30,105

Don't

105,30

60,60

Figure 24. A Fully Reduced Marriage Game

The Schelling point Nash equilibrium is for each player to take the vow and invest, and thus the payoff that will occur if a vow can be taken is 110, 110, the "efficient" outcome. In effect, willingness to take the vow is a "signal" that the partner intends to invest in the relationship -- if (s)he didn't, it would make more sense for him (her) to avoid the vow. Both partners are better off if the vow is taken, and if they had no opportunity to bind themselves with a vow, they could not attain the blissful outcome at the upper left.

Thus, when each partner decides whether or not to take the vow, each rationally expects a payoff of 110 if the vow is taken and 70 if not, and so, the rational thing to do is to take the vow. Of course, this depends strictly on the credibility of the commitment. In a world in which marriage vows become of questionable credibility, this reasoning breaks down, and we are back at Table 5, the Prisoners' Dilemma of "investment in the relationship." Some sort of first-stage commitment is necessary. Perhaps emotional commitment will be enough to make the partnership permanent -- emotional commitment is one of the things that is missing from this "rational" example. But emotional commitment is hard to judge. One of the things a credible vow does is to signal emotional commitment. If there are no vows that bind, how can emotional commitment be signaled? That seems to be one of the hard problems of living in modern society!

There is a lot of common sense here that your mother might have told you -- anyway my mother would have! What the game-theoretic analysis gives us is an insight on why Mom was right, after all, and how superficial reasoning can mislead us. If we compare the payoffs in Figure 24, we can observe that, given the investment choices made, no-one is ever better off in the upper proper subgame (vow) than in the lower (no vow). And except for the invest, invest strategies, both parties are worse off with the vow than without it. Thus I might reason -- wrongly! -- that since, ceteris paribus, I am better off with freedom of choice than without it, I had best not take the vow. But this illustrates a pitfall of "ceteris paribus" reasoning. In this comparison, ceteris are not paribus. Rather, the outcomes of the various subgames -- "ceteris" -- depend on the payoff possibilities as a whole. The vow changes the whole set of payoff possibilities in such a way that "ceteris" are changed -- non paribus -- and the outcome improved. The set of possible outcomes is worse but the selection of outcomes among the available set is so much improved that both parties are almost twice as well off as they would be had they not agreed to restrain their freedom of choice.

In other words: Cent' Anni!