Chapter 1:
Introduction

What's a Game?

Every child understands what a game is. However, mathematical games, which are the content of this online book have a slightly different meaning. So we will have to describe clearly the ingredients of a mathematical game.

Rules: Games have rules. These rules specify what is allowed, what isn't. Though many real-world games allow creativity of finding out new "moves", new ways to act, those games that can be analyzed mathematically have a very rigid set of possible moves, usually all of them known in advance to all involved. But keep in mind---in real world decisions, you always have one more option than you are aware of!

It's not just a game: Games usually are characterized by the lack of seriousness. However, it was the purpose of game theory, from its beginning in 1928, to be useful also to serious situations in economics, politics, business, and others. Even war, although never been called a game, can be analyzed by mathematical game theory.

Outcome and payoffs: Children (and also many grown-ups) play for hours just for the fun of it. Maybe this feature, being just a lot of fun but without use or result, would be the definition of the word "game" in everyday language. In contrast to this, a mathematical game must have an outcome. Moreover, each such outcome has payoffs for the different players attached, which may be monetary or just expressing the satisfaction of that outcome for that player. You want to win money, or at least be declared the winner.

Uncertainty of the Outcome: Games should have a "thrill" insofar as the outcome cannot be predicted in advance. Given that the rules are usually fixed, this implies that a game either must involve some randomness or more than one player.

Make decisions: A game where you don't make decisions might be boring, at least for the mind. Running 100 meter as fast as you can doesn't require a mathematician, or even your brain---it just requires fast legs. However, most sport games also involve decisions, and can therefore, at least partly analyzed by game theory. Some other games at least require the decision whether to play or not.

Cheating: Almost every game knows cheating. Cheating means not playing by the rules, doing something which is not considered a possible move. When your chess opponent is distracted, you take your queen and put it on a better field. Or in poker, you exchange your "8" in your hand with an Ace in your sleeve. Note that game theory doesn't acknowledge the existence of cheating. We will learn how to win without cheating.

Game, Play, Move: Some Definitions

The complete set of rules describes a game. This should be distinguished from a play, which is a certain instance of the game. In certain situations, called positions, a player has do make a decision. This decision is called a move or an action. This should not be confused with strategies. A strategy is a plan that tells the player what move to choose in each possible position where the player has to move.

Rational Behavior is usually assumed for all players. It means that every player has well-defined preferences, has beliefs about the world (including the other players) and tries the best to maximize his or her individual satisfaction. Moreover, it also means that every player is aware that all other players are trying to maximize their satisfaction, and that they are aware that he or she is trying to maximize his, and that he or she is aware that they are ... and so on.

Classification of Games

Games can be categorized under several categories:

• Number of Players: The most obvious distinction of games is by the number of players involved. Usually there should be more than one player. However, you can play Roulette alone---the casino doesn't count as player since it doesn't make any decisions. It takes or gives the money, if necessary. Most monographs on game theory don't include one-player games, but in this text I will allow them provided they contain elements of randomness.
• Simultaneous, sequential, and other games: In a simultaneous game, each player has only one move, and all these moves are done simultaneously. In a sequential game, no two players move at the same time, and players may have to move several times. Obviously there are games that are neither simultaneous no sequential.
• Randomness: Games may contain some random events which would influence the outcome of the game. These events are called "random moves", executed by the so-called "random player".
• Perfect information: A sequential game has perfect information if every player, when about to move, knows all moves done by the other players (including the random player) so far. Since this is impossible in simultaneous moves, the notion only applies to sequential games.
• Complete information means that all players know the structure of the game---the order in which the different players move, all possible moves in each situation, and the payoffs for the different players for all possible outcomes. Although these requirements often fail in real-world games, we assume complete information in most cases, since games of incomplete information are rather difficult to analyze..
• Zero-sum Games have the property that for every possible outcome of the game the sum of the returns of all players equals zero. Then a player can only win if some other loses. Poker, chess, are examples of zero-sum games. Real-world situations are rarely zero-sum.
• Communication: Sometimes communication is allowed before and between the moves, sometimes it is not.
• Cooperative versus non-cooperative games: Even if communication is allowed, the real question is whether the results of these negotiations can be enforced. If not, a player can always move different to what he or she promised in the negotiation. Then this communication is called "cheap talk". A cooperative game is one where the results of the negotiations can be put into a contract, and where there is some institution enforcing these contracts. Moreover there must be some means of distributing the won payoff between the members of the coalition after the game.

We will start investigating simultaneous games.

References

• John Von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior, Princeton University Press 1944.
• Game Theory A survey over the history with a lot of literature.
• History of Game Theory
• Game Theory, Stanford Encyclopedia of Philosophy

Exercises

1. In English Auction, an item is auctioned. People increase bets in increments of $10, and the player giving the highest bet gets the item for that amount of money. Describe reasons why the auctioneer would be considered a player of the game, or reasons why he would not considered to be a player. Does the game contain random moves? Is it a zero-sum game or not? Discuss whether a real-world art auction would have complete information or not.
2. Consider the casino game Roulette. Would the croupier considered to be a player or not? Does the game contain random moves? Is it zero-sum or not? Can the outcome of roulette be improved if some players form a coalition and discuss how to play before each round?
3. Look at the game ROCK-SCISSORS-PAPER. How many players are there? Is it simultaneous, or sequential, or neither, and if it is sequential, does it have perfect information?
4. Discuss number of players, whether it is sequential or simultaneous, or neither, and if it is sequential, whether it has perfect information for poker. Discuss whether there are random moves. Is communication allowed in poker?
5. Discuss number of players, whether it is sequential or simultaneous, or neither, and if it is sequential, whether it has perfect information for Black Jack. Discuss whether there are random moves. Is communication allowed in Black Jack?
6. It's late afternoon and you are sitting in a train going along a coastline. From time to time the train stops in villages, some of them nice, some of them ugly, but what's important is that you can evaluate the niceness of the village immediately. Moreover, the benefit of the evening and night spent at that night depends only on the niceness of the village. For this reason you want to wait for the nicest village before you leave. Unfortunately you don't know how many villages are still to come, and you know nothing about how villages in this country look "normally". What makes things worse is that you are not able to ask anybody, since you don't speak the language of the country.. what niceness to expect. You also know that some (unknown) time in the evening the train will reach its final destination---then you have to stay there whether it is nice or not. Explain the different features of this game, with emphasis on the informational issues. How would you play it? Give some reason for your strategy.
   (Initially I formulated this example in terms of marriage in a catholic society, where divorce is impossible, but have been convinced that this is a different game. Could you give some arguments why?)
   Comment on whether we have complete or incomplete information here, and why.
7. In this slightly more realistic version of the game above you know that overall the train will stop in 10 villages before it reaches its terminal destination. How would you play now?
   Comment on whether we have complete or incomplete information here, and why.