Game Theory

Haha. Got nothing better to do so was doing some reading online. And it just so happens I got to this site about the game theory that we recently studied on. So just wanted to post it up so the people resting in Penang, Melaka, Johor and KL don't slack off (yes... I'm talking about u FETG-1ians). The site featured the Prisoner's Dilemma game.

Like we we did in da economic chaps, the situation is more clearly defined by: 'two players are partners in a crime who have been captured by the police. Each suspect is placed in a separate cell, and offered the opportunity to confess to the crime.'

And then we have the ever-so headacheable Payoff Matrix:

	not confess	confess
not confess	5,5	-4,10
confess	10,-4	1,1

Now, as we already know, the higher the numbers, the higher the utility (or freedom merits). So, if both prisoners do not confess to the crime, alas, they both go free - thus the 5,5 equilibrium. However, there is a catch (as repeated over and over again by Winnie and Cheng You cuz I never went to class.. xD) - if one of the two prisoners actually confess, they will be selling the other person out. The confessor gets the entire 10 utility whereas the one who did not confess gets all the blame, thus the -4 util. If both of them confess, however, they will both be given a reduced term in prison (better something than nothing at all).

Now, lets leave morales and personal intentions aside. It has already been known that this game relates heavily with economic theorists, for several reasons:

1st of all. It can be substituted with a variety of important situations, for example using 'for common good' or for 'personal gain' instead of the Prisoner Dilemma game. This can be associated with basic economic problems, for example, building a bridge. It is best for everyone if the bridge is built, but it is best for an individual if someone else did the building for them. It can also be used to describe two firms competing in the same market (ok... this is the part we've been talking about so many times before), by labelling the dimensions as 'set low price' and 'set high price'. Now, while it is best for consumers if both the firms set a low price, but, it will be best for both individual firms if the other sets a relatively higher price in orer to sacrifice some of thier market share for the other party's gain.

Secondly, it is self-evident as to how rational a person should be. it doesn't matter what the suspect expects his partner to do (referring back to the PD game), it is always better to confess. Why? Because of the odds. If his partner confesses, he gets 10 instead of 5. If his partner doesn't confess, he gets 1 instead of -4. Then if you both confess, its a sure dead-drop 1. The chances of getting either 10 or 1 is deliberately more sensible than risking a -4 because you 'believe' in your partner. This becomes the common question that befalls many economic problems about individual and public interests.

Thirdly, is that it will change significantly if it is repeated, or if the players ever meet each other again in the future. Now, lets say the first game is over, and they are both set free or released from jail after serving their terms. Lets say now they committed another crime and were brought back to the same situation, and then round 2 is played. This time, both suspects will ought to surrender and confess. They might reason in the first round that the partner will not sell them out. But after round 1, the prisoners already know what their partner is like, or what they will do again if they were brought back into the same situation. So, this repetition makes it possible for new rewards or punishments.

Now, I got this part here right off the net, by David K. Levine, and its called the Pride Game. Its kinda more complicated as he brings in the 'pride' factor into the game theory. But this is really interesting.

If We Were All Better People The World Would Be A Better Place

Let us start with a variation on the Prisoner's Dilemma game we may call the Pride Game.

	proud	not confess	confess
proud	4.0, 4.0	5.4, 3.6	1.2, 0.0
not confess	3.6, 5.4	5.0, 5.0	-4.0, 10.0
confess	0.0, 1.2	10.0, -4.0	1.0, 1.0

The Pride Game is like the Prisoner's Dilemma game with the addition of the new strategy of being proud. A proud individual is one who will not confess except in retaliation against a rat-like opponent who confesses. In other words, if I stand proud and you confess, I get 1.2, because we have both confessed and I can stand proud before your humiliation, but you get 0, because you stand humiliated before my pride. On the other hand, if we are both proud, then neither of us will confess, however, our pride comes at a cost, as we both try to humiliate the other, so we each get 4, rather than the higher value of 5 we would get if we simply chose not to confess. It would be worse, of course, for me to lose face before your pride by choosing not to confess. In this case, I would get 3.6 instead of 4, and you, proud in the face of my humiliation would get 5.4.

The Pride Game is very different than the Prisoner's Dilemma game. Suppose that we are both proud. In the face of your pride, if I simply chose not to confess I would lose face, and my utility would decline from 4 to 3.6. To confess would be even worse as you would retaliate by confessing, and I would be humiliated as well, winding up with 0. In other words, if we are both proud, and we each believe the other is proud, then we are each making the correct choice. Morever, as we are both correct, anything either of us learns will simply confirm our already correct beliefs. This type of situation - where players play the best they can given their beliefs, and they have learned all there is to learn about their opponents' play is called by game theorists a Nash Equilibrium.

Notice that the original equilibrium of the Prisoner's Dilemma confess-confess is not an equilibrium of the Pride game: if I think you are going to confess, I would prefer to stand proud and humiliate you rather than simply confessing myself.

Now suppose that we become "better people." To give this precise meaning take this to mean that we care more about each other, that is, we are more altruistic, more generous. Specifically, let us imagine that because I am more generous and care more about you, I place a value both on the utility I receive in the "selfish" game described above and on the utility received by you. Not being completely altruistic, I place twice as much weight on my own utility as I do on yours. So, for example, if in the original game I get 3 units of utility, and you get 6 units of utility, then in the new game in which I am an altruist, I get a weighted average of my utility and your utility. I get 2/3 of the 3 units of utility that belonged to me in the original "selfish" game, and 1/3 of the 6 units of utility that belonged to you in the "selfish" game. Overall I get 4 units of utility instead of 3. Because I have become a better more generous person, I am happy that you are getting 6 units of utility, and so this raises my own utility from the selfish level of 3 to the higher level of 4. The new game with altruistic players is described by taking a weighted average of each player's utility with that of his opponent, placing 2/3 weight on his own utility and 1/3 weight on his opponent's. This gives the payoff matrix of the Altruistic Pride Game

	proud	not confess	confess
proud	4.00, 4.00	4.8, 4.20*	0.80, 0.40
not confess	4.20*, 4.80	5.00, 5.00	0.67, 5.33*
confess	0.40, 0.80	5.33*, 0.67	1.00*, 1.00*

What happens? If you are proud, I should choose not to confess: if I were to be proud I get a utility of 4, while if I choose not to confess I get 4.2, and of course if I do confess I get only 0.4. Looking at the original game, it would be better for society at large if when you are proud I were to choose not to confess. This avoids the confrontation of two proud people, although of course, at my expense. However, as an altruist, I recognize that the cost to me is small (I lose only 0.4 units of utility) while the benefit to you is great (you gain 1.4 units of utility), and so I prefer to "not confess." This is shown in the payoff matrix by placing an asterisk next to the payoff 4.2 in the proud column.

What should I do if you choose not to confess? If I am proud, I get 4.8, if I choose not to confess I get 5, but if I confess, I get 5.33. So I should confess. Again, this is marked with an asterisk. Finally, if you confess, then I no longer wish to stand proud, recognizing that gaining 0.2 by humiliating you comes at a cost of 1 to you. If I choose not to confess I get only 0.67. So it is best for me to confess as well.

What do we conclude? It is no longer an equilibrium for us both to be proud. Each of us in the face of the other's pride would wish to switch to not confessing. Of course it is also not an equilibrium for us both to choose not to confess: each of us would wish to switch to confessing. The only equilibrium is the box marked with two asterisks where we are both playing the best we can given the other player's play: it is where we both choose to confess. So far from making us better off, when we both become more altruist and more caring about one another, instead of both getting a relatively high utility of 4, the equilibrium is disrupted, and we wind up in a situation in which we both get a utility of only 1. Notice how we can give a precise meaning to the "world being a better place." If we both receive a utility of 1 rather than both receiving a utility of 4, the world is clearly a worse place.

The key to game theory and to understanding why better people may make the world a worse place is to understand the delicate balance of equilibrium. It is true that if we simply become more caring and nothing else happens the world will at least be no worse. However: if we become more caring we will wish to change how we behave. As this example shows, when we both try to do this at the same time, the end result may make us all worse off.

To put this in the context of day-to-day life: if we were all more altruistic we would choose to forgive and forget more criminal behavior. The behavior of criminals has a complication. More altruistic criminals would choose to commit fewer crimes. However, as crime is not punished so severely, they would be inclined to commit more crimes. If in the balance more crimes are committed, the world could certainly be a worse place. The example shows how this might work.

Now I know its really confusing, but the conclusion is this - if we choose to actually think about another person's needs and how they feel, then yeah... the world will be a better place.... and its a proven GAME theory.

PS: Copyright of David K. Levine, Department of Economics, UCLA, taken and edited from levine.sscnet.ucla.edu