Chapter 19: Games As Game Theory Systems

Chapter 19: Games as Game Theory Systems These are the main problems: How does each player plan his course—i.e., how does one formulate an exact concept of a strategy? What information is available to each player at every stage of the game? What is the role of a player being informed about the other player's strategy? About the entire theory of the game?—Oscar Morganstern and John Von Neumann, Theory of Games and Economic Behavior Introducing Game Theory? Perhaps you thought that this entire book was about game theory. If that were the case, what does "Games as Game Theory Systems" mean? Actually,game theory is not what it may appear to be. It is not a general term that means theoretical approaches to games. Game theory means something quite specific: it is a branch of economics that can be traced back to the work of two mathematicians, Oscar Morganstern and John Von Neumann. The classic text in the field is Theory of Games and Economic Behavior, published in 1942. Game theory is the mathematical study of decision making. It looks at how people behave in specific circumstances that resemble very simple kinds of games. The founders of game theory intended to create a new kind of mathematical approach to the study of economics. Morganstern and Von Neumann were writing during a time when Marxism was very much in vogue in the field of economics, and Theory of Games and Economic Behavior was, in many ways, an attempt to replace the ideological approach of Marxism with a more rational and scientific set of techniques. Although it caused quite a sensation when it was introduced, the promises of game theory were never quite fulfilled, and it has largely fallen out of favor as a methodology within economics. But game theory can still be quite useful for game designers. The theory of games is a theory of decision making. It concerns how one should make decisions and, to a lesser extent, how one does make them.You make a number of decisions every day. Some involve deep thought, while others are almost automatic.Your decisions are linked to your goals—if you know the consequences of each of your options, the solution is easy. Decide where you want to be and choose the path that takes you there.When you enter an elevator with a particular floor in mind (your goal), you push the button (one of your choices) that corresponds to your floor.[1] As a formal game design schema, Games as Game Theory Systems looks at games as systems of rational choice. It is potentially useful to game designers for two chief reasons. Chapter 19: Games as Game Theory Systems 1 2 Chapter 19: Games as Game Theory Systems First, it analyzes situations that resemble simple games in a very detailed way. Even more importantly, as game theorist Morton D. Davis points out in the previous quotation, game theory specifically focuses on relationships between decisions and outcomes. We know from our earlier discussion of interactivity that actions and outcomes are the building blocks of meaningful play. Within this schema, we explore questions of how players plan their course of action within a game and how they formulate strategies and make decisions. From decision trees to degenerate strategies, we will look closely at the application of game theory concepts to the design of meaningful play. [1]Morton D. Davis, Game Theory: A Nontechnical Introduction (Mineola: Dover Publications, 1970), p. 3. < Day Day Up > < Day Day Up > Decision Trees As a formal approach to understanding games, game theory looks at games as a series of strategic decisions made by the players of a game. What does it mean to reduce a game to its strategic decisions? One common game theory method is to create a decision tree for a game. A decision tree is a branching tree-style diagram that outlines all of the possible moves a player can make in a game. Decision trees are a common way of flow-charting interactive experiences. For example, if you are programming an interactive story that has a hypertext structure, you might draw a diagram that shows all of the links between the different parts of your story. This kind of diagram would be a decision tree. Creating a decision tree for a game is more complicated than creating a decision tree of a hypertext structure. The difference is that in a typical hypertext, the links and the actions that a player can perform at any location in the larger structure do not change as the participant moves through the structure. A reader's first choice in a hypertext structure does not change the way that the other hypertext links function. The only thing that changes is where the reader is positioned in the structure. A game is more complex. In a game, what you can do at any given moment depends on what has already happened in the game. At the beginning of a game of Chess, for example, you can't move either of your rooks because they are both blocked by pawns. Later in the game, you might be able to move your rooks if your pawns have been maneuvered out of the way. The complexity of games leads to a system of many possible actions. Which actions can happen at a given moment is contingent on the current state of the game. Because Chess is a complicated game to diagram as a decision tree, let's start with a simpler example: Tic-Tac-Toe. In Prisoner's Dilemma, a book about game theory and its historical context, writer William 2 Chapter 19: Games as Game Theory Systems Chapter 19: Games as Game Theory Systems 3 Poundstone leads us through the process of making a decision tree of the game of Tic-Tac-Toe. Ticktacktoe starts with the first player ("X") putting a mark in any of nine cells. There are consequently nine possible first moves. The nine choices open to Player X on the first move can be diagrammed as nine lines radiating up from a point. The point represents the move, the moment of decision, and the lines represent the possible choices. Next it's Player O's move. There are eight cells still open—which eight depending on where the X is. So draw eight secondary branches at the top of each of the nine primary branches. That leaves seven open cells for X on his second move. As the diagram of possible moves is continued upward, it branches like a very bushy tree. As you continue the process, you will eventually diagram moves that put three markers in a row. That's a win for the player who moves. It's also the termination of that particular branch in the diagram, for the game ends when someone gets three in a row. Mark that point (call it a "leaf" of the diagram) as a win for X or O as the case may be. Other branches of the diagram will terminate in a tie. Mark them as ties. Obviously, the game of ticktacktoe cannot go on forever. Nine moves is the maximum. So eventually, you will have a complete diagram of the game of ticktacktoe. Every possible ticktacktoe game—every game that ever has been played or ever will be played—must appear in the diagram as a branch starting at the "root"(X's first move) and continuing up to a "leaf" marked as a win for X, a win for O, or a tie. The longest complete branches/games are nine moves long. The shortest are five moves (this is the minimum for a win by the first player).[2] Creating a decision tree can be a powerful way of understanding the formal structure of a game. It is in essence a way of mapping out a game's formal space of possibility. For a simple game such as Tic-Tac-Toe, the complete space of possibility can in fact be diagrammed. However, not all games can be mapped out in this way. Being able to make a decision tree of a game or other interactive structure implies that the decisions participants make are discrete decisions that lead to knowable outcomes. For example, a game that involves physical skill, such as American Football, does not have self-contained moments of decision making that can be diagrammed like the alternate turn-taking of Tic-Tac-Toe. Instead, the game exists as a continuous flow of action. When the ball is hiked, a quarterback does not take a single discrete action. Instead, the game flows forward in a complex web of activity. Perception, movement, and the granularity of the real world creates a non-discrete game space. Although the moment-to-moment play of Football is continuous, the game can be broken down into a system of separate plays. Does that mean it is possible to create a decision tree of Football by widening the frame of analysis, so that each decision point on the chart represents the choice of a play by one team's coach? The answer is no. The problem with this proposal is that to create a decision tree, the result of a decision needs to be a knowable outcome or set of outcomes. Think about a game of Tic-Tac-Toe. When a player makes a decision to place an X or an O in a particular square, there isn't any doubt that the player will finish the action and make the mark. On the other hand, just because a Football team picks a certain play does not mean that they will be able to successfully complete it, or complete it in a way that can be predicted with any accuracy.

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