Game Theory I
Posted in Economics on March 28th, 2010 by admin – Comments OffA close friend directed me to the Yale open course series on Game Theory, and I’ve decided to work through it in addition to the ridiculously heavy courseload I have this term. In order to keep things straight, I’m going to try to do one lesson per week and post my notes, as well as my personal thoughts, online.
If anyone else is taking the time to go through this stuff, feel free to drop in your ideas and suggestions in the comments. Just so everyone is aware, my thoughts will be in blockquotes, as such:
This is a personal thought.
Without further ado:
Introduction/Definition
Strategic situations require multiple players each of which has some affect on the outcomes of the situation. In situations where there is only a single player making decisions, or situations in which players have only one “right” action, there is no strategic requirement (and hence, no game).
Game One
To begin with, let’s consider a game in which you and an another anonymous player must set a grade “bid.” In this scenario, each of you has exactly two choices, alpha and beta. The outcome of the game will be dependent upon the choices each player makes. If both players choose alpha, then both players will receive a B-. Should both players select beta, then both players receive a B+. Finally, should one player choose alpha, and the other beta, the alpha player will receive an A, while the beta player will receive a C.
We can show this visually by constructing the following two tables:

In order to make the game information easier to comprehend, we can take the information from table 2, and superimpose it over table 1, as such:

In this table, we can easily see that P1 (presumably us) is the row player, while P2 is the column player. In order to develop a strategy, however, we need to assign some sort of value to each grade. To do this, we will start by assuming that both players only care about their own grade – in short, both players are what we will call “assholes”. By adding numerical values, we come up with the following table:

Each of the above numbers represents a point value for the payoff. So, given the above matrix, we can state that player 1’s optimal strategy is to choose alpha in every case. this is because no matter what player 2 does, choosing alpha always results in a net gain as 0 points is better than -1, and 3 points is better than 1. When a particularly choice offers superior payouts in all instances, we say that it is a dominating strategy. In this case, beta would be a strictly dominated strategy, as no matter what player 2 does, player 1’s payout is worse in every case.
This leads us to our first lesson of game theory, primarily never play a strictly dominated strategy.
Now, before we go further, consider for a moment that if both players have the same realization about the payoffs, then both players will always play alpha, resulting in both players getting 0 points every time. This is still worse than if both players chose beta every time – which would allow each to get a single point. However, from the point of view of any single player in this game, choosing beta is a bad idea, as the opposition player then has a strong incentive to choose alpha, thus maximizing their own reward and diminishing yours. This leads us to another lesson: Rational choice (playing the dominant strategy) can lead to inefficient outcomes.
I’ve tried to bring this up in economics classes before, namely to do with competition and trade theory. Part of the issue is that rational self-interest in competitive markets creates a sub-optimal equilibrium.
Let’s look at a slightly different situation, however, in which both players are “indignant angels.” In this example, our angels feel guilty about getting an “A” at the expense of their classmates, and so the value of 3 is modified by a -4. Alternately, in the event that our angel gets a “C”, this person is even more distraught at the injustice – hence, indignant. This creates the following payout matrix:

It’s important to note that there is no dominant strategy in this case. By changing the payoffs, we now have no clear means of choosing a course of action. Do we minimize losses or maximize gains? This is a coordination problem. It also leads us to our third lesson: payoffs matter – so unless you know what you want, you cannot get what you want. In other words; know what your goals are.
Lastly, we need to consider what happens when our asshole player from the first example, plays against our indignant angel from the second example. By combing our payoff matrices, we end up with the following table:

As in the first case, alpha is still the dominating strategy. As a result, player one should always choose to play alpha. However, if the roles are reversed, and we are player 2, something very interesting happens. While player 2 still has no dominant strategy, if P2 knows P1’s payoff matrix, then P2 knows that P1 will always choose alpha. Therefore, P2’s best strategy is to always choose alpha as well, since to do otherwise will guarantee a loss.
This leads us to our fourth lesson of the class: Knowing what others will do and understanding other people’s payoffs gives you a powerful advantage in determining strategy.


