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Game Theory Games
Game theory demands a sacred character for rules of behavior which may not be observed in reality. The real world, with all its emotional, ethical, and social suasions, is a far more muddled skein than the Hobbesian universe of the game theorist.—Richard Epstein, The Theory of Gambling and Statistical Logic
Now that we have outlined decision trees and strategies, we are ready to take a look at what it is that game theory calls a game. As the mathematician Richard Epstein points out, game theory games are not about real-world situations or about all kinds of games. Game theorists look at very particular kinds of situations in a very narrow way. What kind of situations? We can summarize a game theory game in the following way: a game theory game consists of rational players who simultaneously reveal a strategy to arrive at an outcome that can be defined in a strict measure of utility. Usually, game theory limits itself to games with only two players.
Rational play, simultaneity, strategy, outcome, utility, and two players. Let us look at each of these elements separately. First, game theory focuses its attention on rational players. Rational players are perfectly logical players that know everything there is to know about a game situation. Furthermore, rational players play to win. As Poundstone puts it, "Perfectly rational players would never miss a jump in checkers or 'fall into a trap' in Chess. All legal sequences of moves are implicit in the rules of these games, and a perfectly logical player gives due consideration to every possibility."[5] As we know from our detailed investigation of Tic-Tac-Toe, if two rational players played the game, the outcome will always end in a draw, because both players would select strategies that would stalemate the other player. Rational players are a fiction, of course, as Epstein makes clear. Real-world players are not like game theory players, as rational as Mr. Spock, completely immune from "emotional, ethical, and social" liabilities. But rational players are still a useful theoretical construct, for they allow us to look at games in a very isolated and controlled way. The fact that rational players follow a strategy is an important aspect of a game theory game as well. As we mentioned previously, a strategy is comprehensive. It is a complete plan for playing an entire game, from start to finish. A strategy includes explicit instructions for playing against any other strategy an opponent selects. In a game theory game, both rational players simultaneously choose and reveal their strategies to each other. In other words, instead of the "I take my turn, you take your turn" pattern of many games, in a game theory game, players only make one decision, at the same time, without knowing what the other player will do. In making a simultaneous decision, a player has to take into account not just the current state of the game, but also what the opponent is thinking at that very moment. A classic example of a simultaneous decision game is Rock-Paper-Scissors, in which both players have to decide what they are going to do based on the anticipated action of the other player. So although game theory does not study psychology directly, there is a psychological element in game theory games, where players might consider "bluffing" or using other indirect strategies against each other. Though they might take these kinds of actions, rational players are still psychologically predictable. Players in a game theory scenario are never going to be vindictive, forgetful, self-destructive, or lazy, as this would change their status as rational players. In game theory games one can always assume that both rational players are acting in their own best interest and are developing strategies accordingly. Why would game theory choose blind, simultaneous decision making as the game play process that it studies? Remember that game theory is not a form of game design: it is a school of economic theory. Within an economic situation, decisions have to be made without knowledge of how the other "players" are going to act. Should you sell your stock in Disney, or buy more shares? Should you purchase two gallons of milk this week, or buy one and wait to see if the price goes down? Should a nation increase or decrease import taxes? All of these micro-and macro-economic scenarios involve making decisions. But the outcome of the decision is based on factors outside the decision maker's direct control. Simultaneous, blind decisions offer a way of simulating this decision making context, a context that lies at the intersection of mathematics and psychology. As Morganstern and Von Neumann explain, It is possible to describe and discuss mathematically human actions in which the main emphasis lies on the psychological side. In the present case, the psychological element was brought in by the necessity of analyzing decisions, the information on the basis of which they are taken, and the interrelatedness of such sets of information (at the various moves) with each other. [6]
Another important component of a game theory game is utility, which is a mathematical measure of player satisfaction. In order to make a formal theory of decision making, it was necessary that Von Neumann and Morganstern numerically quantify the desire of a player to achieve a certain outcome. In a game theory game, for every kind of outcome that a decision might have, a utility is assigned to that decision. A utility function is simply a "quantification" of a person's preferences with respect to certain objects. Suppose I am concerned with three pieces of fruit: an orange, an apple, and a pear. The utility function first associates with each piece of fruit a number that reflects its attractiveness. If the pear was desired most and the apple least, the utility of the pear would be greatest and the apple's utility would be least.[7]
Utility can become more complex when multiple factors come into play. For example, if you were building a house for yourself on beachfront property, thinking in game theory terms, you could measure different locations of your house in terms of utility. You might be able to get the highest utility, say +10, if you built right on the beach. There might be a lower utility, such as +5 or +2, if you had to build it several meters away from the shoreline. On the other hand, if you had to build the house so far away from the beach that the ocean was no longer in view, your utility might go into the negative numbers, indicating an outcome that you would find unpleasant. Of course, you might not have the money to afford the situation with the highest utility. For example, you might require a house of a certain size and if it were directly on the beach it couldn't have a basement and would have to be smaller. Or the cost of the house might be higher on the beach because of the extra architectural complexity required to build in the sand. Cost, size, and location would all be assigned different values. In making your decision, you would try and maximize the total utility given your available options. These examples touch on the ways that game theory employs the concept of utility. It might seem silly to turn something like human satisfaction into a numerical value, given the innumerable complexities that go into our feelings of pleasure, but Morganstern and Von Neumann felt very strongly that a scientific theory of economics necessitated such an approach. In their book, they use an analogy to physical properties such as heat. Before scientists developed a way of conceptualizing and measuring heat, it was an unknown, fuzzy property that seemed impossible to measure: a sensation that occurred as one approached a flame. But the precise measurement of heat is now an important part of contemporary physics. The aim of Von Neumann and Morganstern was to begin a similar revolution in economics, by quantifying pleasure as a measure of utility. Utility may well be an oversimplification of human desire, but it does make a good fit with the formal qualities of games. As we know from our definition of games, all games have a quantifiable outcome: someone wins, or loses, everyone wins or loses, or player performance is measured in points, time, or some other numerical value. The concept of assigning a numerical utility to decision outcomes is really just another way of creating a quantifiable outcome. When looking at games through a formal frame, we do not have the luxury of being non-numerical. The formal systems of both digital and non-digital games require an exactness that does in fact come down to numbers. How many kills did you earn that round? What qualifying time do you need on the next heat in order to continue the race? Which team won the game? These very simple game results are all quantifiable outcomes, and are all examples of utility as well. The last component of most game theory games is that they are usually played by only two players. This was not part of the original formulation of game theory as proposed in Theory of Games and Economic Behavior. The original idea was that the theory could cover n-player games, where n was a number of any size that indicated the number of players. But Von Neumann and Morganstern found that, as with the problem of three planetary bodies discussed in Games as Emergent Systems, their theory became vastly more complex when it took three or more players into account. As a result, most game theory work has focused on two player games. We follow suit in the material to follow. [5]Richard Epstein, The Theory of Gambling and Statistical Logic (San Diego: Academic Press, 1995), p. 118.[6]John Von Neumann and Oscar Morganstern, Theory of Games and Economic Behavior (Princeton: Princeton University Press, 1944), p. 77.[7]Davis, Game Theory: A Nontechnical Introduction, p. 62.
