A Game Theory Approach on PHP

PHP stands for (in Bahasa) “Pemberi Harapan Palsu”, or “The Giver of False Hope”. It signifies the situation where someone (say, Adam) gives attention/love to other person (say, Britney) in hope that the other person will reciprocate his love back. If Britney decides not to reciprocate back, or sending mixed signal (thus either deliberately/unknowingly “fish” Adam to give Britney more love), Britney can be said as a PHP.

The setup is like this: let’s say the initial amount of love (l) for Adam and Britney is 5 for each person. Adam can either not give a damn to Britney (payoff is 5, 5 for both of them) or he can give that love to her. If Adam decides to give all his “love stock” to B, it is being multiplied (m) to be 20 for Britney, then Britney can decide whether she wants to split that 20 “love stock” for both of them (payoff is 10, 10) or keep them for herself (payoff is 0, 20 for Adam and Britney subsequently) thus resulting in Adam’s “love stock” to become zero, a.k.a. he is gonna be devastated or heart-broken. Zing! Britney can also decide to give only Adam portion of attention (keeping a portion of love stock x to herself), let’s say, 6 “love stock”, resulting in a payoff of (6, 14) for Adam and Britney.

The movements here are represented with g, n, r, k which stand for “give love”, “not give a damn”, “reciprocate”, and “keep”, respectively. Probability (p) of Britney to either play r or k is arbitrarily chosen as 50%, though in real life you can’t assign definite probability to love. Also, in this illustration, r means a 50:50 split of the “love stock” (not including the x, see above paragraph), which is also an arbitrary condition. END means one of the parties decides to stop the game.

The setup, assuming Britney will always either split in half or not at all, with 50% probability.

The usual Nash equilibrium strategy for Adam and Britney is (n, k) - Adam must never play the game; Britney, whenever Adam plays the game, must always keep all the love for herself and leave Adam with nothing. This will give a payout of (5, 20) for them. However, this strategy is not an evolutionary stable strategy if the game is iterated. Not to mention, you know, love is sorta continuous, amirite?.

Adam can rationally decide to give love continuously (in N-times iterated/repeated games) if he can be *100% sure* that Σ (m . li – xi), from i = 1 to N, will always ≥ N . l. But, this is not always the case, as Adam can never control Britney’s mind to reciprocate his love, hence there exist variable p and x which may evolve (and can't be rationally predicted by Adam) over times. Therefore Adam’s continuous love is irrational as certainty isn’t assured. Using coin to randomly determine Britney's responses in repeated games, you can see how Adam is basically being stupid if he decides to play along.

Table of Adam's payoffs. Adam:  stupid person.

Britney’s position, on the other hand, will always put her in advantageous place no matter what Adam is doing. However, in order to maximize her payoffs, she needs to “fish” Adam into playing the game repeatedly. She can mix her “reciprocate” and “keep” signal to give Adam an “illusion of love” to keep playing the game. She won’t get lose anyway. Hence: the PHP.

In order to maximize his position, Adam must keep track of Britney’s reactions, and leave the game immediately whenever Britney decides not to respond to his love. But, in real life, this is not the case, why? Economically speaking, Adam unconsciously shifts his reference point to zero whenever he plays each section of these repeated games of love. Therefore, in his mind, it’s as if the game is always starting from zero (no accumulated losses, no accumulated gains) and discontinuous.

 

Adam: not knowing that he is stupid.

Genuine love is irrational. Being a PHP is rational.

Q.E.D.



P.S.:

There are some caveats. Note that: 

a) The coin that I used to randomly determine Britney's responses may or may not be a fair coin (putting aside the Central Limit Theorem and law of large number.), 

b) Using coin to determine Britney's responses was, strictly speaking, misapplied, since I was not intending to portray Britney as if she uses mixed strategy at the very first place.

Even though Adam's expected value Σ pi (m . li - xi) from i = 1 to N on the second row of the table on the ilustration above equals to N . l, that is, both are 50, Adam's decision not to give a damn bears much better result than if he chooses to give love to Britney. The act of giving love is a costly signalling, though, I didin't incorporate such signal costs in the illustration aforementioned.