Here’s the game:
Leyla and William both want to be in a relationship, but disagree whether it should be an emotional or a non-emotional one.
Leyla wants to have an emotional/commited relationship. William wants to have a non-emotional/non-commited relationship. Each player gets a utility of 2 if both players get into his/her type of relationship, a utility of 1 if both get into the other’s type of relationship, and 0 if they are unable to agree and break up or start dating someone else.
This game has two Nash equilibriums in pure strategy with payoffs (2,1) and (1,2). And one mixed equilibrium with payoffs (0,0).
Leyla will play emotional with probability 2/3 and non-emotional with probability 1/3.
William will play non-emotional with probability 2/3 and emotional with probability 1/3.
This is simple, just take x, y to be the probabilities that Leyla plays emotional and William plays non-emotional respectively. Williams’s indifference between emotional and non-emotional is equivalent to:
0* non-emotional + 2*(1- non-emotional)=1* non-emotional + 0*(1- non-emotional)
Or
non-emotional =2/3.
Similarly for Leyla, to be indifferent between emotional and non-emotional, it must be that
emotional =2/3.
The problem with this game is that there is no unique solution and either of the Nash equilibriums will frustrate one of the players. If players have not played this game before, it is hard to see just what the right prediction might be because there is no way for the players to coordinate their expectations. These two Nash equilibriums: (1,2) and (2,1) are actually Pareto efficient (Pareto efficiency means that you cannot make anyone happier my changing strategy keeping everything else constant), but in real life they will cause frustration to both players and will not lead to a healthy long-term relationship.
However, things are not as gloomy as one might think. Schelling, an economist, devised a theory of “focal points” in the 60s, suggesting that in real-life situations, players might be able to coordinate on a particular equilibrium by using information that is abstracted away by the strategic form. For example:
1. Additional information about the future
2. Players’ culture
3. Past experiences.
However, one shouldn’t be optimistic too soon because there are still several other points to consider:
1. The Nash equilibrias ((1,2) and (2,1)) are unfair because one player consistently does better than the other.
Leyla will play emotional with probability 2/3 and non-emotional with probability 1/3.
William will play non-emotional with probability 2/3 and emotional with probability 1/3.
This is simple, just take x, y to be the probabilities that Leyla plays emotional and William plays non-emotional respectively. Williams’s indifference between emotional and non-emotional is equivalent to:
0* non-emotional + 2*(1- non-emotional)=1* non-emotional + 0*(1- non-emotional)
Or
non-emotional =2/3.
Similarly for Leyla, to be indifferent between emotional and non-emotional, it must be that
emotional =2/3.
The problem with this game is that there is no unique solution and either of the Nash equilibriums will frustrate one of the players. If players have not played this game before, it is hard to see just what the right prediction might be because there is no way for the players to coordinate their expectations. These two Nash equilibriums: (1,2) and (2,1) are actually Pareto efficient (Pareto efficiency means that you cannot make anyone happier my changing strategy keeping everything else constant), but in real life they will cause frustration to both players and will not lead to a healthy long-term relationship.
However, things are not as gloomy as one might think. Schelling, an economist, devised a theory of “focal points” in the 60s, suggesting that in real-life situations, players might be able to coordinate on a particular equilibrium by using information that is abstracted away by the strategic form. For example:
1. Additional information about the future
2. Players’ culture
3. Past experiences.
However, one shouldn’t be optimistic too soon because there are still several other points to consider:
1. The Nash equilibrias ((1,2) and (2,1)) are unfair because one player consistently does better than the other.
2. (0,0) mixed strategy is inefficient because neither player gets any utility.
3. Time matters. Players must make the decision at the same time. Otherwise, if one player moves first, the other will have to select the first player’s move to ensure he/she gets any utility at all. Outcome here will be (1,2) or (2,1) depending on who plays first.
4. Is having something better than having nothing? An economist would say "Yes". If you were given the choice to get 1 cent or get nothing, you should bother to get the 1 cent as long as the costs of getting the cent are smaller than the value of the 1 cent coin. But the costs of being in a type of relationship you don't like (frustration, disappointment) outweight the benefits of being in a relationship in the first place. Love is never enough.
5. There is the problem of irrationality in real-life situations. Humans are not “homo economicus”, but “homo irrationalus”. One player might chose to hurt the other player by making the final outcome be (0,0), where neither player wins. This is the worst of all.
In game theory, this game is called the “unburned battle-of-the-sexes”. There is one way to make this game work. Transform it into a “burned battle-of-the-sexes”, that is, allowing players to destroy some of their utility. This is very applicable to real life situations: if the player’s don’t reach an equilibrium, players lose. Thus, the player’s utility is, say, -2, and in that case an equilibrium ensues. In real-life, the player risks to leave broken-hearted, so one player gives up and the couple moves in together or gets married. I have to think about whether this is actually feasible, but I don’t think I like this solution. A commited relationship can't come out of fear, but out of pleasure. It's like having sex because you enjoy it, not because you fear you won't get it if you don't do it then.
The other way to make it work, is Schelling’s strategy: cues and communication (verbal, non-verbal, etc). Now, get William talking and you have conquered the world!
