Games Zone

Some of the power and meaning of game theory can be illustrated by assessing the statement "If we were all better people the world would be a better place." This may seem to you to be self-evidentally true. Or you may recognize that as a matter of logic this involves the fallacy of composition: just because a statement applies to each individual person it need not apply to the group. Game theory can give precise meaning to the statement of both what it means to be better people and what it means for the world to be a better place, and so makes it possible to prove or disprove the statement. In fact the statement is false, and this can be shown by a variation of the Prisoner's Dilemma.

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.

For those of you who are interested in or already know more advanced game theory, the Pride Game has only the one Nash equilibrium shown - it is solvable by iterated strict dominance. The Atruistic Pride Game, however, has several mixed strategy equilibria. You can compute them using the fine open source software program Gambit written by Richard McKelvey, Andrew McLennan and Theodore Turocy. One equilibrium involves randomizing between proud and confess, so is worse than the proud-proud equilibrium of the Pride game. The other is strictly mixed in that it randomizes between all three strategies. The payoffs to that equilibrium gives each player 2.31 - so while it is better than both players confessing for certain, it is still less good than the unique equilibrium of the Pride Game.

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I'd like to thank Jie Zheng for his help. The "We are all better people" example is inspired by and based on an academic paper by Sung-Ha Hwang and Samuel Bowles called "Is Altruism Bad For Cooperation?" If you know some basic calculus the paper is very readable, and I will provide a link as soon as they post it online. They provide a much more persuasive and robust example of how altruism hurts cooperation, tightly linked to experimental evidence. If you wish to learn more about game theory, there a variety of good books on the topic.

One way to describe a game is by listing the players (or individuals) participating in the game, and for each player, listing the alternative choices (called actions or strategies) available to that player. In the case of a two-player game, the actions of the first player form the rows, and the actions of the second player the columns, of a matrix. The entries in the matrix are two numbers representing the utility or payoff to the first and second player respectively. A very famous game is the Prisoner's Dilemma game. In this game the 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. The game can be represented by the following matrix of payoffs

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

Note that higher numbers are better (more utility). If neither suspect confesses, they go free, and split the proceeds of their crime which we represent by 5 units of utility for each suspect. However, if one prisoner confesses and the other does not, the prisoner who confesses testifies against the other in exchange for going free and gets the entire 10 units of utility, while the prisoner who did not confess goes to prison and which results in the low utility of -4. If both prisoners confess, then both are given a reduced term, but both are convicted, which we represent by giving each 1 unit of utility: better than having the other prisoner confess, but not so good as going free.

This game has fascinated game theorists for a variety of reasons. First, it is a simple representation of a variety of important situations. For example, instead of confess/not confess we could label the strategies "contribute to the common good" or "behave selfishly." This captures a variety of situations economists describe as public goods problems. An example is the construction of a bridge. It is best for everyone if the bridge is built, but best for each individual if someone else builds the bridge. This is sometimes refered to in economics as an externality. Similarly this game could describe the alternative of two firms competing in the same market, and instead of confess/not confess we could label the strategies "set a high price" and "set a low price." Naturally it is best for both firms if they both set high prices, but best for each individual firm to set a low price while the opposition sets a high price.

A second feature of this game, is that it is self-evident how an intelligent individual should behave. No matter what a suspect believes his partner is going to do, it is always best to confess. If the partner in the other cell is not confessing, it is possible to get 10 instead of 5. If the partner in the other cell is confessing, it is possible to get 1 instead of -4. Yet the pursuit of individually sensible behavior results in each player getting only 1 unit of utility, much less than the 5 units each that they would get if neither confessed. This conflict between the pursuit of individual goals and the common good is at the heart of many game theoretic problems.

A third feature of this game is that it changes in a very significant way if the game is repeated, or if the players will interact with each other again in the future. Suppose for example that after this game is over, and the suspects either are freed or are released from jail they will commit another crime and the game will be played again. In this case in the first period the suspects may reason that they should not confess because if they do not their partner will not confess in the second game. Strictly speaking, this conclusion is not valid, since in the second game both suspects will confess no matter what happened in the first game. However, repetition opens up the possibility of being rewarded or punished in the future for current behavior, and game theorists have provided a number of theories to explain the obvious intuition that if the game is repeated often enough, the suspects ought to cooperate.

What economists call game theory psychologists call the theory of social situations, which is an accurate description of what game theory is about. Although game theory is relevant to parlor games such as poker or bridge, most research in game theory focuses on how groups of people interact. There are two main branches of game theory: cooperative and noncooperative game theory. Noncooperative game theory deals largely with how intelligent individuals interact with one another in an effort to achieve their own goals. That is the branch of game theory I will discuss here.

In addition to game theory, economic theory has three other main branches: decision theory, general equilibrium theory and mechanism design theory. All are closely connected to game theory.

Decision theory can be viewed as a theory of one person games, or a game of a single player against nature. The focus is on preferences and the formation of beliefs. The most widely used form of decision theory argues that preferences among risky alternatives can be described by the maximization of the expected value of a numerical utility function, where utility may depend on a number of things, but in situations of interest to economists often depends on money income. Probability theory is heavily used in order to represent the uncertainty of outcomes, and Bayes Law is frequently used to model the way in which new information is used to revise beliefs. Decision theory is often used in the form of decision analysis, which shows how best to acquire information before making a decision.

General equilibrium theory can be viewed as a specialized branch of game theory that deals with trade and production, and typically with a relatively large number of individual consumers and producers. It is widely used in the macroeconomic analysis of broad based economic policies such as monetary or tax policy, in finance to analyze stock markets, to study interest and exchange rates and other prices. In recent years, political economy has emerged as a combination of general equilibrium theory and game theory in which the private sector of the economy is modeled by general equilibrium theory, while voting behavior and the incentive of governments is analyzed using game theory. Issues studied include tax policy, trade policy, and the role of international trade agreements such as the European Union.

Mechanism design theory differs from game theory in that game theory takes the rules of the game as given, while mechanism design theory asks about the consequences of different types of rules. Naturally this relies heavily on game theory. Questions addressed by mechanism design theory include the design of compensation and wage agreements that effectively spread risk while maintaining incentives, and the design of auctions to maximize revenue, or achieve other goals.

Games are objects which consist of components and rules and have certain criteria: rules, a goal, always changing course; chance; competition; common experience; equality; freedom; activity; diving into the world of the game; and no impact on reality.

For judging what is or is not a game, the basic criteria, which not only apply to "games with rules", but to all games, is as important as the special criteria for "games with rules". Here are the criteria which all games have.

• Common experience
• Equality
• Freedom
• Activity
• Diving into the world of the game

4.1 Common experience

Games bring people together, regardless of gender, generation, and race. Most games are multi-player games which lead to group experiences, which linger after the game is over. But there are groups of games which are played alone. These are the so-called solitaire games and most computer games.

4.2 Equality

In a game, all players are equal and have the same chance to win. Where else in this does absolute equality exist? I think that is one of the reasons that children love to play games, because in a game with adults, they are equal partners.

4.3 Freedom

Whoever plays a game, does it from his freedom of choice. He is not forced or coerced by anyone to play. Playing games is not work, not commitment, nothing you have to do. Therefore, we can say that playing games means being free. This freedom is basic to all games. Here the embracing game term has its value.

4.4 Playing means being active

Whoever reads a book, watches a movie, or listens to music, consumes or acquires, but does not act. While nowadays most leisure activities seduce people into passivity, the game makes people act. Depending on the game, the following activities may be undertaken:

Spiritual Area

• thinking, combining
• planning
• making decisions
• concentrating
• training your mind
• receiving knowledge
• understand the impact of systems

Emotional Area

• rules, accepting laws
• to learn how to work with others
• to learn how to lose
• to learn more about yourself and others
• to use fantasy and creativity

Motor Area

• practice skillfulness
• practice reactions

I am sure that games do not fulfill a purpose, but are not useless. Activity is basic to all games. Here again, the embracing game term has its value.

4.5 Diving into the world of the game

Whoever plays, leaves behind reality and dives into the world of the game. These game worlds are comparable to reality. This statement still applies to the big difference between the two worlds. I want to substantiate that the course of reality and the course of the game world will be steered by the same factors:

• laws/rules (natural laws and human laws - game rules, which are like laws)
• chance
• your own doing (within the frame of predetermined laws)
• competition (achievement of the best)
• course and end are unknown same language and means of expression

Despite these similarities, the game world is not the everyday world and reality is not a game world. Game worlds have a limited space and have a different understanding of time.

A game is only a game, when everything that happens in the game stays within the game world. This is not the case when the outcome influences reality immensely. In this case, the game is reality, for example games of chance (Roulette, Poker, Black Jack) or with professional sports activities (Rugby, Baseball, ice hockey, Cricket, Olympic games, chess championships and so on). With all reality games, the principal of freedom gets lost. With the reality games, you play the game from commitment not freedom. The game results directly effect reality.

Games are entertainment and fun. The players may only take their emotions from the game world into reality. Whoever spends an interesting and fun evening playing games with friends, takes this happy feeling into reality. By crossing this small border between the real world and the game world has the result that during the game one is relaxed and can escape from the real world and then return relaxed and happy.

There are criteria which apply to all games and those that apply to the "games with rules". First, I will describe the criteria that apply to "games with rules".

• Game rules
• Goal
• The course of the game is never the same - chance
• Competition

3.1 Game rules

As already discussed, the rules and the components define the game. Everything that is in the rules is part of the game. Everything that is not in the rules does not belong in the game. The rules are the borders and the heart of the game. They only refer to the game and never exist outside of the game. Although the game has rules which are like laws, playing a game is voluntary and cannot be forced on the players. Whoever plays a game, voluntarily binds himself to the rules. Where force is involved, there is no game. All games without rules are not "games with rules".

3.2 Goal

Every game has a goal. Thus, there are two definitions:

• The victory condition or requirement.
• The strategy needed to win the game.

I would like to make clear the difference between the two definitions with an example. In the game Go, the victory condition is to earn the most points. In order to achieve this, a player must win space. Thus, the strategy, which players use during the game, is to win space. Therefore, I define the game goal as the strategy, which the players work on to win.

There are thousands of games, but only a small number of game goals. That means that most games have the same game goal. At first this seems surprising. But when we look at it closely and see that every has a winner and a loser, the goal of the game must be something measurable, relatively simple to measure, and depicted in a game.

3.3 The course of the game is never the same - chance

This attribute, of all entertainment media, is only found in a game. Someone who reads a book, watches a movie, or listens to music, can repeat the experience at any time, but the course and the content is always the same. You can play a game any number of times, however, and the course will always be different. Also, with each game, the course is unknown and it is uncertain who will win the game. Uncertainty and unknown, that is what make games so exciting and delightful. The reason for this is in the game rules and the chance, which play a larger or smaller role in each game.

Playing is experimenting with chance (Novalis). Chance will be experienced in a game by luck (or bad luck). Games, which are mostly based on chance, offer little development possibilities for a player and are usually boring.

On the other hand, chance makes games unpredictable and interesting, and causes the game's course to develop differently each time. How does chance get into a game:

• with a random generator (e.g. dice)
• with different start-up situations (e.g. dealing cards)
• with incomplete information (e.g. moving at the same time, unknown strategy of your fellow players)
• with a very high number of move options

Pure strategy games have some chance elements. If that were not so, the game's course would be too deterministic, and we wouldn't like a game whose result was known at the beginning. In strategy games, chance is shown in the large number of possible moves. Because of the many moves, no player knows the winning strategy, which leads to victory.

All games which have the same course, by definition, do not belong to "games with rules". For example, this would be puzzles, quizlets, and brain teasers, which lose their attraction when they have been solved. Solitaire games which follow a different course each time belongs, for me, to "games with rules". For example the card game Patience. Very interesting and informative in this connection is the Games Cafe article What Is a Puzzle? by Scott Kim.

3.4 Competition

Each game demonstrates competition. Players compete in a game. There are winners and losers. Even in cooperative games or when players work as a team, competition exists. In this case, the players compete against one of the predetermined situations, that means the players play cooperatively against the game system. The same applies to solitaire games (e.g. Patience, Solitaire). A competition needs a system, in which the game results can be compared. The competition and the measurement of the game results are criteria which limit the game and the cause that certain feelings won't be fulfilled throughout the game as in books, movies, and music. For example, love, freedom, harmony, pain, sorrow, etc. The criteria "competition" is also the reason why it is so difficult to develop games which are not aggressive. Please see the Games Journal article Non Predatory Games by Kate Jones.

In most games, the rules are more significant than the components. But there are games where these roles are reversed: where the components are significant and the rules not very important at all. Usually, these are action games like Looping Louie.

The components are the hardware, the rules are the software. Both define the game. Both can exist independently from each other, but separately are not a game.

Archeology finds ancient game boards and game pieces, but no one knows what rules these ancients used to play their games. We will never know how these games were played.

Components and rules can be combined:

• a set of components may be used with different rules.
• a set of rules can be used with different components.

Suppose we just had the rules for Halma, but not the board and pieces and had to reconstruct the game.

• What should the board look like?
• How many spaces should it have?
• What shape do the spaces have: square, hexagonal, or round?
• How are the spaces laid out?
• Are all the spaces the same size or are they of different sizes?
• How many pieces are there?
• What do the pieces look like? Does a piece take more than one space when played?

The rules are not sufficient to define a game! (Unless the rules have pictures of the components and game situations.)

In the German language a game is any activity which is executed only for pleasure and without conscious purpose. In this definition every activity that brings pleasure is a game. For example, people dance, play musical instruments, act in plays, and play with dolls and model trains.

This definition people use today comes from the works of Johan Huizinga (Homo Ludens, 1938) and Friedrich Georg Jünger (Die Spiele, 1959). But there are more ways to define games. Manfred Eigen's and Ruthild Winkler's definition for game goes beyond the definition used by Huizinga. They see a game as a natural phenomenon: half necessity and half coincidence (Das Spiel, 1975). Their definition of games comes closer to Adornos' definition, who set himself apart from Huizinga by identifying games as an art form.

But in our sense these definitions are too wide, we define game more succinctly. Thus, I am writing about games which belong to the class that includes Chess, 9 Man's Morris, Checkers, Halma, Go, Parchisi, Monopoly, Scrabble, Skat, Rummy, Bridge, Memory, Jack Straws, Dominoes, and so on. Unfortunately, our language does not have a good term to call these games. Terms like table games, society games, tournament games are too narrow. In my opinion, the best term would be "rulegames" = "games with rules".

I will now try to find the criteria for defining "game with rules". Kevin Maroney defined game in his Games Journal article, My Entire Waking Life. Scott Kim defined puzzle in his Games Cafe article "What Is a Puzzle?" as separate from a game. Kate Jones writes about less aggressive games in her Games Journal article Non Predatory Games. My definition is a further attempt to explore the nature of games.