And apparently, intrinsically worthless gifts (e.g. an engagement ring?) are great signals.

THE SETUP:

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.

POSSIBLE EQUILIBRIA:

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?

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.

MORE PLAYERS?

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!

In English, Incentives make you want to do something you otherwise wouldn’t want to do. Today let’s talk about an incentive which is in the media at the moment, the oft criticised, proposed marriage tax break.

The plan is essentially an easy one. Cut the costs for people who want to get married via a tax rebate, and it makes sense. There is a lot of evidence to support the claim that marriage is good for society.

As the Telegraph reports

The statistical evidence is overwhelming: children brought up by two parents do much better, on average, than children brought up by just one. They are less likely to drop out of school or end up in prison, and more likely to pay taxes and remain in employment. This is not to deny that single parents do an excellent job and raise children who grow up to contribute much to society. It is simply to point out that across the population as a whole, two-parent families are more likely to achieve that result – and couples who marry are five times more likely to stay together, and provide that positive environment, than those who cohabit without any public commitment.

We all benefit from well-adjusted children, who are ‘polite, well educated and contribute to society’ this is what economics calls an ‘externality’ a benefit by a party that is not directly involved in the transaction. Parents are paying an extra  cost which we all benefit from.

Well the first is the claim by Ed Balls that the tax is “trying to socially engineer family life through a tax policy… [it] is hugely expensive and unfair.”. An economist would claim however that “Ahhhh! All taxes change peoples behaviour”  and The Working Families Tax Credit increased the benefits available to single mothers who sought employment thus penalising women who stayed home to look after their children or who remained with their husbands.(the latter claim being VERY  VERY contentious)

The second claim is that the credits “could stigmatise children” whose parents are not married. Again many feel that this is false by providing tax incentives for marriage it is difficult to see how it would  “stigmatise” the children of the unmarried, any more than providing incentives for single parenthood would “stigmatise” those whose parents live together.

There is of course one issue with the above arguments, they all involve children,  supporting and subsidising ALL married couples regardless of their parental status, in effect , penalises everyone else. So should the ‘tax break’ be administered to just  married couples with children? I don’t know?  Of course, nothing will replace a loving, stable, family; interested in their child’s upbringing and this is something that no taxbreak can fix.