Game theoretical models

Even a quite brave soldier may prefer to run rather than heroically, but pointlessly, die trying to stem the oncoming tide all by himself. Thus we could imagine, without contradiction, a circumstance in which an army, all of whose members are brave, flees at top speed before the enemy makes a move.

What we have here, then, is a case in which the interaction of many individually rational decision-making processes—one process per soldier—produces an outcome intended by no one.

Most armies try to avoid this problem just as Cortez did. During the Battle of Agincourt Henry decided to slaughter his French prisoners, in full view of the enemy and to the surprise of his subordinates, who describe the action as being out of moral character.

The reasons Henry gives allude to non-strategic considerations: he is afraid that the prisoners may free themselves and threaten his position. However, a game theorist might have furnished him with supplementary strategic and similarly prudential, though perhaps not moral justification. His own troops observe that the prisoners have been killed, and observe that the enemy has observed this. Metaphorically, but very effectively, their boats have been burnt. The slaughter of the prisoners plausibly sent a signal to the soldiers of both sides, thereby changing their incentives in ways that favoured English prospects for victory.

These examples might seem to be relevant only for those who find themselves in sordid situations of cut-throat competition. Perhaps, one might think, it is important for generals, politicians, mafiosi, sports coaches and others whose jobs involve strategic manipulation of others, but the philosopher should only deplore its amorality. Such a conclusion would be highly premature, however.

The study of the logic that governs the interrelationships amongst incentives, strategic interactions and outcomes has been fundamental in modern political philosophy, since centuries before anyone had an explicit name for this sort of logic. Philosophers share with social scientists the need to be able to represent and systematically model not only what they think people normatively ought to do, but what they often actually do in interactive situations.

The best situation for all people is one in which each is free to do as she pleases. Often, such free people will wish to cooperate with one another in order to carry out projects that would be impossible for an individual acting alone.

But if there are any immoral or amoral agents around, they will notice that their interests might at least sometimes be best served by getting the benefits from cooperation and not returning them. Suppose, for example, that you agree to help me build my house in return for my promise to help you build yours.

After my house is finished, I can make your labour free to me simply by reneging on my promise. I then realize, however, that if this leaves you with no house, you will have an incentive to take mine.

This will put me in constant fear of you, and force me to spend valuable time and resources guarding myself against you. I can best minimize these costs by striking first and killing you at the first opportunity. Of course, you can anticipate all of this reasoning by me, and so have good reason to try to beat me to the punch. Since I can anticipate this reasoning by you , my original fear of you was not paranoid; nor was yours of me.

In fact, neither of us actually needs to be immoral to get this chain of mutual reasoning going; we need only think that there is some possibility that the other might try to cheat on bargains. Once a small wedge of doubt enters any one mind, the incentive induced by fear of the consequences of being preempted —hit before hitting first—quickly becomes overwhelming on both sides. If either of us has any resources of our own that the other might want, this murderous logic can take hold long before we are so silly as to imagine that we could ever actually get as far as making deals to help one another build houses in the first place.

The people can hire an agent—a government—whose job is to punish anyone who breaks any promise. So long as the threatened punishment is sufficiently dire then the cost of reneging on promises will exceed the cost of keeping them.

The logic here is identical to that used by an army when it threatens to shoot deserters. If all people know that these incentives hold for most others, then cooperation will not only be possible, but can be the expected norm, so that the war of all against all becomes a general peace.

Few contemporary political theorists think that the particular steps by which Hobbes reasons his way to this conclusion are both sound and valid. Working through these issues here, however, would carry us away from our topic into details of contractarian political philosophy. What is important in the present context is that these details, as they are in fact pursued in contemporary debates, involve sophisticated interpretation of the issues using the resources of modern game theory.

Notice that Hobbes has not argued that tyranny is a desirable thing in itself. The structure of his argument is that the logic of strategic interaction leaves only two general political outcomes possible: tyranny and anarchy. Sensible agents then choose tyranny as the lesser of two evils.

The distinction between acting parametrically on a passive world and acting non-parametrically on a world that tries to act in anticipation of these actions is fundamental. The values of all of these variables are independent of your plans and intentions, since the rock has no interests of its own and takes no actions to attempt to assist or thwart you.

Furthermore, his probable responses should be expected to visit costs upon you, which you would be wise to consider. Finally, the relative probabilities of his responses will depend on his expectations about your probable responses to his responses.

The logical issues associated with the second sort of situation kicking the person as opposed to the rock are typically much more complicated, as a simple hypothetical example will illustrate. Suppose first that you wish to cross a river that is spanned by three bridges. Assume that swimming, wading or boating across are impossible.

The first bridge is known to be safe and free of obstacles; if you try to cross there, you will succeed. The second bridge lies beneath a cliff from which large rocks sometimes fall. The third is inhabited by deadly cobras. Now suppose you wish to rank-order the three bridges with respect to their preferability as crossing-points. The first bridge is obviously best, since it is safest. To rank-order the other two bridges, you require information about their relative levels of danger. Your reasoning here is strictly parametric because neither the rocks nor the cobras are trying to influence your actions, by, for example, concealing their typical patterns of behaviour because they know you are studying them.

It is obvious what you should do here: cross at the safe bridge. Now let us complicate the situation a bit. Your decision-making situation here is slightly more complicated, but it is still strictly parametric. However, this is all you must decide, and your probability of a successful crossing is entirely up to you; the environment is not interested in your plans. However, if we now complicate the situation by adding a non-parametric element, it becomes more challenging.

Suppose that you are a fugitive of some sort, and waiting on the other side of the river with a gun is your pursuer. She will catch and shoot you, let us suppose, only if she waits at the bridge you try to cross; otherwise, you will escape.

As you reason through your choice of bridge, it occurs to you that she is over there trying to anticipate your reasoning.

It will seem that, surely, choosing the safe bridge straight away would be a mistake, since that is just where she will expect you, and your chances of death rise to certainty. So perhaps you should risk the rocks, since these odds are much better. But wait … if you can reach this conclusion, your pursuer, who is just as rational and well-informed as you are, can anticipate that you will reach it, and will be waiting for you if you evade the rocks.

So perhaps you must take your chances with the cobras; that is what she must least expect. But, then, no … if she expects that you will expect that she will least expect this, then she will most expect it. This dilemma, you realize with dread, is general: you must do what your pursuer least expects; but whatever you most expect her to least expect is automatically what she will most expect. You appear to be trapped in indecision. All that might console you a bit here is that, on the other side of the river, your pursuer is trapped in exactly the same quandary, unable to decide which bridge to wait at because as soon as she imagines committing to one, she will notice that if she can find a best reason to pick a bridge, you can anticipate that same reason and then avoid her.

We know from experience that, in situations such as this, people do not usually stand and dither in circles forever. However, until the s neither philosophers nor economists knew how to find it mathematically. As a result, economists were forced to treat non-parametric influences as if they were complications on parametric ones. This is likely to strike the reader as odd, since, as our example of the bridge-crossing problem was meant to show, non-parametric features are often fundamental features of decision-making problems.

Classical economists, such as Adam Smith and David Ricardo, were mainly interested in the question of how agents in very large markets—whole nations—could interact so as to bring about maximum monetary wealth for themselves. Economists always recognized that this set of assumptions is purely an idealization for purposes of analysis, not a possible state of affairs anyone could try or should want to try to institutionally establish. But until the mathematics of game theory matured near the end of the s, economists had to hope that the more closely a market approximates perfect competition, the more efficient it will be.

No such hope, however, can be mathematically or logically justified in general; indeed, as a strict generalization the assumption was shown to be false as far back as the s. This article is not about the foundations of economics, but it is important for understanding the origins and scope of game theory to know that perfectly competitive markets have built into them a feature that renders them susceptible to parametric analysis. Because agents face no entry costs to markets, they will open shop in any given market until competition drives all profits to zero.

This implies that if production costs are fixed and demand is exogenous, then agents have no options about how much to produce if they are trying to maximize the differences between their costs and their revenues. These production levels can be determined separately for each agent, so none need pay attention to what the others are doing; each agent treats her counterparts as passive features of the environment. The other kind of situation to which classical economic analysis can be applied without recourse to game theory is that of a monopoly facing many customers.

However, both perfect and monopolistic competition are very special and unusual market arrangements. Prior to the advent of game theory, therefore, economists were severely limited in the class of circumstances to which they could straightforwardly apply their models.

Philosophers share with economists a professional interest in the conditions and techniques for the maximization of welfare. In addition, philosophers have a special concern with the logical justification of actions, and often actions must be justified by reference to their expected outcomes.

One tradition in moral philosophy, utilitarianism, is based on the idea that all justifiable actions must be justified in this way. Without game theory, both of these problems resist analysis wherever non-parametric aspects are relevant. In doing this, we will need to introduce, define and illustrate the basic elements and techniques of game theory.

An economic agent is, by definition, an entity with preferences. Game theorists, like economists and philosophers studying rational decision-making, describe these by means of an abstract concept called utility. This refers to some ranking, on some specified scale, of the subjective welfare or change in subjective welfare that an agent derives from an object or an event. For example, we might evaluate the relative welfare of countries which we might model as agents for some purposes by reference to their per capita incomes, and we might evaluate the relative welfare of an animal, in the context of predicting and explaining its behavioral dispositions, by reference to its expected evolutionary fitness.

In the case of people, it is most typical in economics and applications of game theory to evaluate their relative welfare by reference to their own implicit or explicit judgments of it. This is why we referred above to subjective welfare. Consider a person who adores the taste of pickles but dislikes onions. She might be said to associate higher utility with states of the world in which, all else being equal, she consumes more pickles and fewer onions than with states in which she consumes more onions and fewer pickles.

However, economists in the early 20th century recognized increasingly clearly that their main interest was in the market property of decreasing marginal demand, regardless of whether that was produced by satiated individual consumers or by some other factors.

In the s this motivation of economists fit comfortably with the dominance of behaviourism and radical empiricism in psychology and in the philosophy of science respectively. Like other tautologies occurring in the foundations of scientific theories, this interlocking recursive system of definitions is useful not in itself, but because it helps to fix our contexts of inquiry.

When such theorists say that agents act so as to maximize their utility, they want this to be part of the definition of what it is to be an agent, not an empirical claim about possible inner states and motivations. Economists and others who interpret game theory in terms of RPT should not think of game theory as in any way an empirical account of the motivations of some flesh-and-blood actors such as actual people.

Rather, they should regard game theory as part of the body of mathematics that is used to model those entities which might or might not literally exist who consistently select elements from mutually exclusive action sets, resulting in patterns of choices, which, allowing for some stochasticity and noise, can be statistically modeled as maximization of utility functions. On this interpretation, game theory could not be refuted by any empirical observations, since it is not an empirical theory in the first place.

Of course, observation and experience could lead someone favoring this interpretation to conclude that game theory is of little help in describing actual human behavior. Some other theorists understand the point of game theory differently. They view game theory as providing an explanatory account of actual human strategic reasoning processes. These two very general ways of thinking about the possible uses of game theory are compatible with the tautological interpretation of utility maximization.

The philosophical difference is not idle from the perspective of the working game theorist, however. As we will see in a later section, those who hope to use game theory to explain strategic reasoning , as opposed to merely strategic behavior , face some special philosophical and practical problems. Since game theory is a technology for formal modeling, we must have a device for thinking of utility maximization in mathematical terms.

Such a device is called a utility function. We will introduce the general idea of a utility function through the special case of an ordinal utility function.

Later, we will encounter utility functions that incorporate more information. Suppose that agent x prefers bundle a to bundle b and bundle b to bundle c. We then map these onto a list of numbers, where the function maps the highest-ranked bundle onto the largest number in the list, the second-highest-ranked bundle onto the next-largest number in the list, and so on, thus:.

The only property mapped by this function is order. The magnitudes of the numbers are irrelevant; that is, it must not be inferred that x gets 3 times as much utility from bundle a as she gets from bundle c. Thus we could represent exactly the same utility function as that above by.

The numbers featuring in an ordinal utility function are thus not measuring any quantity of anything. For the moment, however, we will need only ordinal functions. All situations in which at least one agent can only act to maximize his utility through anticipating either consciously, or just implicitly in his behavior the responses to his actions by one or more other agents is called a game.

Agents involved in games are referred to as players. If all agents have optimal actions regardless of what the others do, as in purely parametric situations or conditions of monopoly or perfect competition see Section 1 above we can model this without appeal to game theory; otherwise, we need it. In literature critical of economics in general, or of the importation of game theory into humanistic disciplines, this kind of rhetoric has increasingly become a magnet for attack.

The reader should note that these two uses of one word within the same discipline are technically unconnected. Furthermore, original RPT has been specified over the years by several different sets of axioms for different modeling purposes. Once we decide to treat rationality as a technical concept, each time we adjust the axioms we effectively modify the concept. Consequently, in any discussion involving economists and philosophers together, we can find ourselves in a situation where different participants use the same word to refer to something different.

For readers new to economics, game theory, decision theory and the philosophy of action, this situation naturally presents a challenge. We might summarize the intuition behind all this as follows: an entity is usefully modeled as an economically rational agent to the extent that it has alternatives, and chooses from amongst these in a way that is motivated, at least more often than not, by what seems best for its purposes.

Economic rationality might in some cases be satisfied by internal computations performed by an agent, and she might or might not be aware of computing or having computed its conditions and implications. In other cases, economic rationality might simply be embodied in behavioral dispositions built by natural, cultural or market selection. Each player in a game faces a choice among two or more possible strategies.

The significance of the italicized phrase here will become clear when we take up some sample games below. A crucial aspect of the specification of a game involves the information that players have when they choose strategies. A board-game of sequential moves in which both players watch all the action and know the rules in common , such as chess, is an instance of such a game. By contrast, the example of the bridge-crossing game from Section 1 above illustrates a game of imperfect information , since the fugitive must choose a bridge to cross without knowing the bridge at which the pursuer has chosen to wait, and the pursuer similarly makes her decision in ignorance of the choices of her quarry.

The difference between games of perfect and of imperfect information is related to though certainly not identical with! Let us begin by distinguishing between sequential-move and simultaneous-move games in terms of information. It is natural, as a first approximation, to think of sequential-move games as being ones in which players choose their strategies one after the other, and of simultaneous-move games as ones in which players choose their strategies at the same time.

For example, if two competing businesses are both planning marketing campaigns, one might commit to its strategy months before the other does; but if neither knows what the other has committed to or will commit to when they make their decisions, this is a simultaneous-move game.

Chess, by contrast, is normally played as a sequential-move game: you see what your opponent has done before choosing your own next action.

Chess can be turned into a simultaneous-move game if the players each call moves on a common board while isolated from one another; but this is a very different game from conventional chess. It was said above that the distinction between sequential-move and simultaneous-move games is not identical to the distinction between perfect-information and imperfect-information games. Explaining why this is so is a good way of establishing full understanding of both sets of concepts.

As simultaneous-move games were characterized in the previous paragraph, it must be true that all simultaneous-move games are games of imperfect information. However, some games may contain mixes of sequential and simultaneous moves. For example, two firms might commit to their marketing strategies independently and in secrecy from one another, but thereafter engage in pricing competition in full view of one another. If the optimal marketing strategies were partially or wholly dependent on what was expected to happen in the subsequent pricing game, then the two stages would need to be analyzed as a single game, in which a stage of sequential play followed a stage of simultaneous play.

Whole games that involve mixed stages of this sort are games of imperfect information, however temporally staged they might be. Games of perfect information as the name implies denote cases where no moves are simultaneous and where no player ever forgets what has gone before.

As previously noted, games of perfect information are the logically simplest sorts of games. This is so because in such games as long as the games are finite, that is, terminate after a known number of actions players and analysts can use a straightforward procedure for predicting outcomes.

A player in such a game chooses her first action by considering each series of responses and counter-responses that will result from each action open to her. She then asks herself which of the available final outcomes brings her the highest utility, and chooses the action that starts the chain leading to this outcome.

This process is called backward induction because the reasoning works backwards from eventual outcomes to present choice problems. There will be much more to be said about backward induction and its properties in a later section when we come to discuss equilibrium and equilibrium selection. For now, it has been described just so we can use it to introduce one of the two types of mathematical objects used to represent games: game trees.

A game tree is an example of what mathematicians call a directed graph. That is, it is a set of connected nodes in which the overall graph has a direction. We can draw trees from the top of the page to the bottom, or from left to right. In the first case, nodes at the top of the page are interpreted as coming earlier in the sequence of actions.

In the case of a tree drawn from left to right, leftward nodes are prior in the sequence to rightward ones. An unlabelled tree has a structure of the following sort:. The point of representing games using trees can best be grasped by visualizing the use of them in supporting backward-induction reasoning.

Just imagine the player or analyst beginning at the end of the tree, where outcomes are displayed, and then working backwards from these, looking for sets of strategies that describe paths leading to them. We will present some examples of this interactive path selection, and detailed techniques for reasoning through these examples, after we have described a situation we can use a tree to model.

Trees are used to represent sequential games, because they show the order in which actions are taken by the players. However, games are sometimes represented on matrices rather than trees. This is the second type of mathematical object used to represent games. For example, it makes sense to display the river-crossing game from Section 1 on a matrix, since in that game both the fugitive and the hunter have just one move each, and each chooses their move in ignorance of what the other has decided to do.

Here, then, is part of the matrix:. Thus, for example, the upper left-hand corner above shows that when the fugitive crosses at the safe bridge and the hunter is waiting there, the fugitive gets a payoff of 0 and the hunter gets a payoff of 1.

Whenever the hunter waits at the bridge chosen by the fugitive, the fugitive is shot. These outcomes all deliver the payoff vector 0, 1. You can find them descending diagonally across the matrix above from the upper left-hand corner.

Whenever the fugitive chooses the safe bridge but the hunter waits at another, the fugitive gets safely across, yielding the payoff vector 1, 0. These two outcomes are shown in the second two cells of the top row. All of the other cells are marked, for now , with question marks. The problem here is that if the fugitive crosses at either the rocky bridge or the cobra bridge, he introduces parametric factors into the game.

In these cases, he takes on some risk of getting killed, and so producing the payoff vector 0, 1 , that is independent of anything the hunter does. In general, a strategic-form game could represent any one of several extensive-form games, so a strategic-form game is best thought of as being a set of extensive-form games.

Where order of play is relevant, the extensive form must be specified or your conclusions will be unreliable. The distinctions described above are difficult to fully grasp if all one has to go on are abstract descriptions.

Suppose that the police have arrested two people whom they know have committed an armed robbery together. Unfortunately, they lack enough admissible evidence to get a jury to convict. They do , however, have enough evidence to send each prisoner away for two years for theft of the getaway car. We can represent the problem faced by both of them on a single matrix that captures the way in which their separate choices interact; this is the strategic form of their game:.

Each cell of the matrix gives the payoffs to both players for each combination of actions. So, if both players confess then they each get a payoff of 2 5 years in prison each. This appears in the upper-left cell. If neither of them confess, they each get a payoff of 3 2 years in prison each. This appears as the lower-right cell. This appears in the upper-right cell. The reverse situation, in which Player II confesses and Player I refuses, appears in the lower-left cell.

Each player evaluates his or her two possible actions here by comparing their personal payoffs in each column, since this shows you which of their actions is preferable, just to themselves, for each possible action by their partner. Player II, meanwhile, evaluates her actions by comparing her payoffs down each row, and she comes to exactly the same conclusion that Player I does.

Wherever one action for a player is superior to her other actions for each possible action by the opponent, we say that the first action strictly dominates the second one. In the PD, then, confessing strictly dominates refusing for both players. Both players know this about each other, thus entirely eliminating any temptation to depart from the strictly dominated path. Thus both players will confess, and both will go to prison for 5 years.

The players, and analysts, can predict this outcome using a mechanical procedure, known as iterated elimination of strictly dominated strategies. Player 1 can see by examining the matrix that his payoffs in each cell of the top row are higher than his payoffs in each corresponding cell of the bottom row.

Therefore, it can never be utility-maximizing for him to play his bottom-row strategy, viz. Now it is obvious that Player II will not refuse to confess, since her payoff from confessing in the two cells that remain is higher than her payoff from refusing.

So, once again, we can delete the one-cell column on the right from the game. We now have only one cell remaining, that corresponding to the outcome brought about by mutual confession. Since the reasoning that led us to delete all other possible outcomes depended at each step only on the premise that both players are economically rational — that is, will choose strategies that lead to higher payoffs over strategies that lead to lower ones—there are strong grounds for viewing joint confession as the solution to the game, the outcome on which its play must converge to the extent that economic rationality correctly models the behavior of the players.

Had we begun by deleting the right-hand column and then deleted the bottom row, we would have arrived at the same solution. One of these respects is that all its rows and columns are either strictly dominated or strictly dominant. In any strategic-form game where this is true, iterated elimination of strictly dominated strategies is guaranteed to yield a unique solution. Later, however, we will see that for many games this condition does not apply, and then our analytic task is less straightforward.

The reader will probably have noticed something disturbing about the outcome of the PD. This is the most important fact about the PD, and its significance for game theory is quite general. For now, however, let us stay with our use of this particular game to illustrate the difference between strategic and extensive forms.

In fact, however, this intuition is misleading and its conclusion is false. If Player I is convinced that his partner will stick to the bargain then he can seize the opportunity to go scot-free by confessing. Of course, he realizes that the same temptation will occur to Player II; but in that case he again wants to make sure he confesses, as this is his only means of avoiding his worst outcome.

But now suppose that the prisoners do not move simultaneously. This is the sort of situation that people who think non-communication important must have in mind. Now Player II will be able to see that Player I has remained steadfast when it comes to her choice, and she need not be concerned about being suckered.

This gives us our opportunity to introduce game-trees and the method of analysis appropriate to them. First, however, here are definitions of some concepts that will be helpful in analyzing game-trees:. Terminal node : any node which, if reached, ends the game. Each terminal node corresponds to an outcome. Strategy : a program instructing a player which action to take at every node in the tree where she could possibly be called on to make a choice.

These quick definitions may not mean very much to you until you follow them being put to use in our analyses of trees below.

It will probably be best if you scroll back and forth between them and the examples as we work through them. Player I is to commit to refusal first, after which Player II will reciprocate when the police ask for her choice. Each node is numbered 1, 2, 3, … , from top to bottom, for ease of reference in discussion. Here, then, is the tree:. Look first at each of the terminal nodes those along the bottom. These represent possible outcomes.

Each of the structures descending from the nodes 1, 2 and 3 respectively is a subgame. If the subgame descending from node 3 is played, then Player II will face a choice between a payoff of 4 and a payoff of 3.

Consult the second number, representing her payoff, in each set at a terminal node descending from node 3. II earns her higher payoff by playing D. We may therefore replace the entire subgame with an assignment of the payoff 0,4 directly to node 3, since this is the outcome that will be realized if the game reaches that node. Now consider the subgame descending from node 2. Here, II faces a choice between a payoff of 2 and one of 0. She obtains her higher payoff, 2, by playing D.

We may therefore assign the payoff 2,2 directly to node 2. Now we move to the subgame descending from node 1. Typically, the individuals are assumed to adopt strategies in accordance with this equilibrium assuming that such an equilibrium exists. For an overview of game theory in marketing, see Moorthy Source: AMA.

Similarly, organization B also has two strategies either to fight for its existence or to cooperate with organization A. In Figure-2, organization A takes the first step that would be followed by organization B later on.

In case, organization A does not enter the market, then its payoffs would be zero. However, if it enters the market, the market situation would be totally dependent on organization B. If they both get into the price war, then both of them would suffer the loss of 3.

On the other hand, if organization B cooperates, then both of them would earn equal profits. In this case, the best option would be that organization A enters the market and organization B cooperates. Simultaneous games are the one in which the move of two players the strategy adopted by two players is simultaneous.

In simultaneous move, players do not have knowledge about the move of other players. On the contrary, sequential games are the one in which players are aware about the moves of players who have already adopted a strategy. However, in sequential games, the players do not have a deep knowledge about the strategies of other players. Simultaneous games are represented in normal form while sequential games are represented in extensive form.

Let us understand the application of simultaneous move games with the help of an example. Suppose organizations X and Y want to minimize their cost by outsourcing their marketing activities. However, they have a fear that outsourcing of marketing activities would result in increase of sale of the other competitor.

The strategies that they can adopt are either to outsource or not to outsource the marketing activities. In Table, it can be seen that both the organizations X and Y are unaware about the strategy of each other.

Both of them work on the perception that the other one would adopt the best strategy for itself. Therefore, both the organizations would adopt the strategy, which is best for them.

The same example can also be used for the explanation of sequential move games. Suppose organization X is the first one to decide whether it should outsource the marketing activities or not.

In Figure-3, the first move is taken by organization X while organization Y would take decision on the basis of the decision taken by X. However, the final outcome depends on the decision of organization Y. In the present case, the second player is aware of the decision of the first player. Constant sum game is the one in which the sum of outcome of all the players remains constant even if the outcomes are different.

Zero sum game is a type of constant sum game in which the sum of outcomes of all players is zero. In zero sum game, the strategies of different players cannot affect the available resources. Moreover, in zero sum game, the gain of one player is always equal to the loss of the other player. On the other hand, non-zero sum game are the games in which sum of the outcomes of all the players is not zero.

A non-zero sum game can be transformed to zero sum game by adding one dummy player. The losses of dummy player are overridden by the net earnings of players. Examples of zero sum games are chess and gambling. In these games, the gain of one player results in the loss of the other player. However, cooperative games are the example of non-zero games. This is because in cooperative games, either every player wins or loses.