So, one week of micro basic game theory revision can drive you to the edge of insanity… Talking to certain people about their love problems has definitely pushed me over that edge. Here is my analysis of love as a dynamic game of imperfect information. Enjoy!
I will base my analysis on a simple two-players model, which can be extended to create love triangles, quadrangles, etc… We have players A and B, who have concave utility functions, and hence are both risk-averse (i.e. they prefer certainty over an uncertain prospect). People are risk-averse to different degrees, and this will affect the payoffs each player faces, and as a result, the way the game is played. In this game, I will assume that both players are very risk-averse (which is true in most cases). Both players are trying to maximise their payoffs.
Firstly nature chooses whether player A will like player B (we have an information set). The subjective probabilities (something B may believe in) that A likes B and A does not like B are p and (1-p) respectively. This probability will affect the final equilibrium as shown later. Player B really fancies player A, but does not know whether player A will reciprocate. Player B has two strategies: profess and not profess his/her love. Player A’s action is to reciprocate or not reciprocate. However, player A will only reciprocate if nature has made him/her like B and vice versa, i.e. A cannot determine whether he/she will reciprocate. Each player’s payoff will depend on where player B is in the game, and what he/she chooses to do. The potential payoffs are as follows:
1) Player A reciprocates and player B professes: A gets 20 and B gets 20 (both players end up happily together, yay!!!).
2) Player A reciprocates and player B does not profess (sad times hah?): A gets 0 (he/she will never find out that he/she could have been a lot happier, but this can not be treated as a loss either) and B gets -10 for being an idiot (a rational fool) and not professing.
3) Player A does not reciprocate and player B professes (the worst thing ever right?): A gets -10 because suddenly he/she is facing an incredibly awkward situation, which clearly causes a lot of distress (like you need to be nice to that other person, explain why you will not reciprocate blah blah blah), B gets -100 for taking the risk whilst being so risk-averse (in other words for being an irrational fool). This is the most embarrassing scenario for both players.
4) Player A does not reciprocate and player B does not professes (a really boring scenario): A gets 0 again for very similar reasons (the lack of knowledge means he/she will never find out that he/she could have been a lot more stressed), B gets -5 for being a rational fool again (he/she will forever question whether A would have reciprocated).
It is probably easier to see the payoffs if you just draw the normal and extensive forms of the game.
Note that there is no strictly dominant strategy in this version of the game due to player B being very risk-averse.
We now look at best responses. It is pointless trying to look for A’s best responses because whether he/she reciprocates is decided by nature. Thus, we look for B’s best responses:
Best response for B given A reciprocates = profess.
Best response for B given A does not reciprocate = not profess.
Thus, we have two equilibria: (reciprocate, profess) = (20,20), (not reciprocate, not profess) = (0,-5). These are not strictly Nash’s equilibria as A’s action is predetermined by nature. Clearly though, the first equilibrium brings more utility to both players.
THE EASIER SOLUTION (i.e. under perfect information):
If we have a third party to provide (signal) player B information about his/her position in the game, then the game is pretty straight forward. Player B, knowing at which node he/she is, will be able to make the best decision for himself/herself. There will be two Nash equilibria. If player B knows that player A will reciprocate, then he/she will profess, making both players happy. On the other hand, if player B knows for sure that player A will not reciprocate, he/she will not profess and avoid the potential embarrassment. This happens sometimes (as I have seen recently), but most of the time people are in the dark about whether the other person like him/her or not.
THE SOLUTION UNDER IMPERFECT INFORMATION:
There are no mixed strategies in this game as player B can profess only once. However, it is probably possible to have mixed strategies if signalling is introduced. Since B knows nothing about A’s feeling, he/she will have to form expected utility from professing and not professing:
E [U(B)/B professes] = 20p – 100(1-p) = 120p – 100
E [U(B)/B does not profess] = -10p – 5(1-p) = -5p – 5
In order for B to profess, the first equality must be greater than the second. Basic calculations give that p must be greater than 0.76. However, remember what I said in the beginning about p being a subjective value and B being very risk-averse, most people will not say that the probability that someone likes them is that high (unless they have better knowledge after interacting).
Signalling in this game can be slightly more complicated as after player B sends a signal, player A will likely send a signal back, and both players will have to form beliefs functions. Sending a signal can incur costs (say buying flowers and presents or trying to look more physically attractive) or no costs (just showing affection). The signal also may or may not increase the probability that player A will reciprocate player B’s feeling. This is probably too difficult for me to analyse right now, so I will leave this for another note in the near (indefinite) future.
We have never done anything like this in the lectures, and my brain is melting… so I will go and write a blues on my piano! But I hope that everyone has seen that love is a really complicated game because people are just rational fools!