Lover's Dilemma

1. Introduction
John Louis von Neumann was a mathematician who got a first degree in chemistry and yet wrote a book on quantum physics and participated in the development of hydrogen bomb, though his most significant contribution in history was of computer science. This paper, however, is about his idea that revolutionized economics [1][2].

Von Neumann once thought that the Cold War was just a simple two person game [3]. That is, the total benefit to the US and USSR in this game, for every combination of strategies, always adds to zero. More specifically, one side benefits only at the expense of the other. (He was wrong in some sense. It was actually a non-zero-sum game that neither side dared to push the button and eventually achieved a win-win situation)


"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
-- John Louis von Neumann

As a scholar, I spent almost 10 years researching the economical behavior of love. I was surprised to discover that the game of love itself is actually economics; and it is not about paying dinner or buying LV for your girlfriends. I thus write this paper as a conclusion of my 10 year research work and present the concepts of the Lover's Dilemma.

Love is a simple two person game. In this game, as in chess or many others, it is assumed that each individual player is trying to maximize his or her own advantage, without concern for the well-being of the other player. The equilibrium for this type of game does not lead to optimums. Even though they may cooperate to achieve a better overall result of the game, they would still choose to act individually. This is the heart of the dilemma.

2. The Dilemma
The Lover's Dilemma is as follows: The lovers, A and B, are in a trouble relationship. It is not necessary to assume that both players in this game are completely selfish and that their only goal is to maximize their own satisfaction. They want the relationship continues without losing their personal pride and emotional dignity. One may ask for break-up, strategically, and thus showing one-sided influence towards the relationship. Nevertheless, such request might end up being fulfilled and they would lose their lover.


Table 1: The Lover's Dilemma in "Win-Win" Terminology

It can be summarized thus: If one asks for break-up and the other begs for continuing the relationship, the beggar loses his pride but is able to get back to his lover. If both agree to continue, they stay together still but lose the chance of showing one-sided influence towards each other. If they both ask for break-up, the relationship ends but they can still maintain their pride and dignity. To most of the players, self satisfaction is more important than the relationship itself. Therefore being a beggar is the worst case which they would try very hard to avoid.

3. Discussion
It is not difficult to realize that this is a non-zero-sum two person game. Specifically, a gain by one player does not necessarily correspond with a loss by another. If only they could both agree to continue the relationship, they would both be better off; however, from a game theorist's point of view, their best play is to request break-up. I am going to discuss the details in this section.

Each player has two options. The outcome of each choice depends on the choice of the other player. However, neither player knows the choice of his or her lover. Even if they were able to talk to each other, neither could be sure that they could trust the other. Assuming the player A is rationally working out his best move. If his partner wants to continue, according to the above table, his best move is to make a strategical break-up request as he then is able to achieve maximum advantage instead of actually ending the relationship. If his partner asks for break-up, his best move is still to break up, as by doing so he receives a relatively better situation than being a beggar. At the same time, player B thinking rationally would also have arrived at the same conclusion and therefore will request for break-up. Thus in a game of love played once by two rational players both will request for ending the relationship.

If reasoned from the perspective of the optimal interest of the group of the couple, the correct outcome would be for both players to continue their relationship, as this would minimize total lost of the group. Any other decision would be worse for the two lovers considered together. However by each following their selfish interests, the players each receive a bad result.

4. Conclusions
If only a player could sacrifice the personal pride and emotional dignity for his or her lover, if only each of them could be sure that the other player would make the same sacrifice, if only they could concern each others, they would both agree to continue their relationship and achieve a better overall result. However, such a sacrifice cannot exist, as it is vulnerable to the treachery of selfish individuals, which we assumed our players to be. Therein lays the true beauty and the maddening paradox of this game of love.

Von Neumann once said: "If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is." I realized, through mathematics, that life is fairly simple, too. At least to many people, the game of love itself is actually as simple as school level calculation.

5. Future Work
I am currently working on the Iterated Lover's Dilemma which means that the game is played repeatedly. Thus each player has an opportunity to "punish" the other player for previous selfish play. Mutual cooperation in the game may then arise as an equilibrium outcome. The incentive to be selfish may then overcome by the threat of punishment, leading to the possibility of a cooperative outcome.

6. Acknowledgements
This work is with help of many people. In particular, I would like to express my greatest gratitude towards my ex girlfriends. Without the invaluable lessons they taught me, this paper would not have been possible. Through the rest of my days, I shall remember their support with greatest appreciation.

7. References
[1] J. von Neumann, "Zur Theorie der Gesellschaftsspiele", Mathematicsche Annalen, Vol. 100, No. 1, Pages 295-320, 1928

[2] J. von Neumann and O. Morgenstern, "Theory of Games and Economic Behavior", Princeton University Press, 1944

[3] W. Poundstone, "Prisoner's Dilemma: John Von Neumann, Game Theory and the Puzzle of the Bomb", Anchor Books, 1993