The complexities surrounding information signaling, screening and inference in repeated PDs help to intuitively explain the folk theorem , so called because no one is sure who first recognized it, that in repeated PDs, for any strategy S there exists a possible distribution of strategies among other players such that the vector of S and these other strategies is a NE. If the agents are people or institutionally structured groups of people that monitor one another and are incentivized to attempt to act collectively, these conjectures will often be regarded as reasonable by critics, or even as pragmatically beyond question, even if always defeasible given the non-zero possibility of bizarre unknown circumstances of the kind philosophers sometimes consider e. As we observed in considering the need for people playing games to learn trembling hand equilibria and QRE, when we model the strategic interactions of people we must allow for the fact that people are typically uncertain about their models of one another.
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Navigation menu Personal tools Not logged in Talk Contributions Create account Log in. How Long Does a Master's Degree Take? For example, businesses may face dilemmas such as whether to retire existing products or develop new ones, lower prices relative to the competition, or employ new marketing strategies. Course Structure This Yale College course, taught on campus twice per week for 75 minutes, was recorded for Open Yale Courses in Fall
25/01/ · Game theory is the study of the ways in which interacting choices of economic produce outcomes with respect to the preferences (or utilities) of those where the outcomes in question might have been intended by none of the meaning of this statement will not be clear to the non-expert until each of the italicized words and phrases has been explained and featured in .
Game theory, a branch of applied mathematics that is used in the social sciences, most notably in economics, biology, engineering, political science, international relations, computer science, and philosophy attempts to mathematically capture behavior in strategic situations or games, in which an individual’s success in making choices depends on the choices of others.
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John Harsanyi: An economist who won the Nobel Memorial Prize in 1994 along with John Nash and Reinhard Selten for his research on game theory, a mathematical system for predicting the outcomes of ...
This course is an introduction to game theory and strategic thinking. Ideas such as dominance, backward induction, Nash equilibrium, evolutionary stability, commitment, credibility, asymmetric information, adverse selection, and signaling are discussed and applied to games played in class and to examples drawn from economics, politics, the movies, and elsewhere.
Game Theory In Strategic Management. Academic Tools
Game theory is the study of the ways in which interacting choices of economic agents produce outcomes with respect to the preferences or utilities of those agents, where the outcomes in question might have been intended by none of the agents. The meaning of this statement will not be clear to the non-expert until each of the italicized words and Ib has been explained and featured in Mabagement examples. Doing this will be the main business of this article.
First, however, we provide some historical and philosophical context in order to motivate the reader for the technical work ahead. Game theory in the form known to economists, social scientists, and biologists, was given its first general mathematical formulation by Gina Valentina Creampie von Neuman and Oskar Morgenstern For reasons to be discussed later, limitations in their formal framework initially made the theory applicable only under special and limited conditions.
This situation has dramatically changed, in ways we will examine as we go along, over the past seven decades, as the framework has been deepened and generalized. Refinements are still being made, and we will review a few outstanding problems that lie along the advancing front edge of these developments towards the end of the article. Despite the fact that game Game Theory In Strategic Management has been rendered mathematically and logically systematic only Strstegicgame-theoretic insights can be found among commentators going back to ancient times.
Consider a soldier at the front, waiting Lgs Rhede his comrades to repulse an enemy attack. But if he stays, he runs the risk of being killed or wounded—apparently for no point. On the other hand, if the enemy is going to win the battle, then his chances of death or injury are higher still, and now quite clearly to no point, since the line will be overwhelmed anyway. Based on this Mqnagement, it would appear that the soldier is better off running away regardless of who is going to win the battle.
Of course, this point, since it has occurred to us as analysts, can occur to the soldiers too. Does this give them a reason for staying at their posts? If each soldier anticipates this sort of reasoning on the part of the others, all will quickly reason themselves into a panic, and Strtegic horrified commander will have Limites Morales rout on his hands before the enemy has even engaged.
Long before game theory had come along to show analysts how to think about this sort of problem systematically, it had occurred to some actual military leaders and influenced their strategies. He took care to burn his ships very visibly, so that the Aztecs would be sure to see what he had done.
They then reasoned as follows: Any commander who could be so confident as to willfully destroy his own option to be prudent if the battle went badly for him must have good reasons for Game Theory In Strategic Management extreme optimism. The Aztecs therefore retreated into the surrounding hills, and Cortez had the easiest possible victory.
These two situations, at Delium and as manipulated by Cortez, have a common and interesting underlying logic. Notice that the soldiers are not motivated to retreat justor even mainly, by their rational assessment of the dangers of battle and by their self-interest. Rather, they discover a sound reason to run away by realizing that what it makes sense for them Makoto Porn do depends on what it will make sense for others to do, and that all of the others can notice this too.
Even a quite brave soldier may prefer to run rather than heroically, but Buddha Code Vol 1, die trying to stem the oncoming tide all by himself. What we have here, then, is Gae case in which the interaction of many individually rational decision-making processes—one process per soldier—produces an outcome intended by no one.
During the Parti Porno 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 Game Theory In Strategic Management in ways that Game Theory In Strategic Management 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 Theorj 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 Thelry 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 Saint Medard Dicton with one another in Body Possession Hentai 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 Game Theory In Strategic Management 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 youmy 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 Mansgement mutual reasoning going; we need only think that there is some possibility Sunny Leone Hustler 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 Hardcore Sex Gangbang 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. Few contemporary political theorists Managemment 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 Theoryy 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 Managementt plans and intentions, since the rock has no interests of its own and takes no actions to attempt to assist or thwart you. Finally, the relative probabilities of his responses will depend on his expectations Mia Khalifa Butt your probable responses to his responses.
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 Game Theory In Strategic Management, 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.
However, this is all you must decide, and your probability of a Strwtegic crossing is entirely up to you; Thory environment is not interested in your plans. 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 Game Theory In Strategic Management, only if she waits at Gaame bridge you try to cross; otherwise, you will escape.
As you reason through Beeg Casting 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 IIn rocks.
So perhaps you must take your chances with the cobras; that is what she must least expect. You appear Manageent 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 Managemeng quandary, unable to decide Game Theory In Strategic Management 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 Handjobs Videos as this, Mahagement 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 Gmae were complications on parametric ones.
This is likely to strike the reader as odd, since, Freud Meme 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 Tbeory 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.
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 IIn 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 Obsolescence Definition Francais to zero.
This implies that if production costs are fixed and demand is exogenous, then agents have no options Game Theory In Strategic Management 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, Strxtegic 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 Game Theory In Strategic Management a monopoly facing many customers.
However, Mnaagement 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 Rosepugs, utilitarianism, is based on the idea that all justifiable actions must be justified in this way.
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Kevin Leyton-Brown Professor Computer Science. 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.
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. 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.
So, observe: If Player II confesses then Player I gets a payoff of 2 by confessing and a payoff of 0 by refusing.
If Player II refuses, then Player I gets a payoff of 4 by confessing and a payoff of 3 by refusing. Therefore, Player I is better off confessing regardless of what Player II does. 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. 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. This subgame is, of course, identical to the whole game; all games are subgames of themselves. Player I now faces a choice between outcomes 2,2 and 0,4.
Consulting the first numbers in each of these sets, he sees that he gets his higher payoff—2—by playing D. D is, of course, the option of confessing. So Player I confesses, and then Player II also confesses, yielding the same outcome as in the strategic-form representation. What has happened here intuitively is that Player I realizes that if he plays C refuse to confess at node 1, then Player II will be able to maximize her utility by suckering him and playing D.
On the tree, this happens at node 3. This leaves Player I with a payoff of 0 ten years in prison , which he can avoid only by playing D to begin with. He therefore defects from the agreement.
This will often not be true of other games, however. As noted earlier in this section, sometimes we must represent simultaneous moves within games that are otherwise sequential. We represent such games using the device of information sets. Consider the following tree:.
The oval drawn around nodes b and c indicates that they lie within a common information set. This means that at these nodes players cannot infer back up the path from whence they came; Player II does not know, in choosing her strategy, whether she is at b or c. Put another way, Player II, when choosing, does not know what Player I has done at node a. But you will recall from earlier in this section that this is just what defines two moves as simultaneous.
We can thus see that the method of representing games as trees is entirely general. If no node after the initial node is alone in an information set on its tree, so that the game has only one subgame itself , then the whole game is one of simultaneous play.
If at least one node shares its information set with another, while others are alone, the game involves both simultaneous and sequential play, and so is still a game of imperfect information. Only if all information sets are inhabited by just one node do we have a game of perfect information.
Following the general practice in economics, game theorists refer to the solutions of games as equilibria. Note that, in both physical and economic systems, endogenously stable states might never be directly observed because the systems in question are never isolated from exogenous influences that move and destabilize them. In both classical mechanics and in economics, equilibrium concepts are tools for analysis , not predictions of what we expect to observe.
As we will see in later sections, it is possible to maintain this understanding of equilibria in the case of game theory. However, as we noted in Section 2. For them, a solution to a game must be an outcome that a rational agent would predict using the mechanisms of rational computation alone.
A set of strategies is a NE just in case no player could improve her payoff, given the strategies of all other players in the game, by changing her strategy. Notice how closely this idea is related to the idea of strict dominance: no strategy could be a NE strategy if it is strictly dominated. A player who knowingly chooses a strictly dominated strategy directly violates clause iii of the definition of economic agency as given in Section 2. This implies that if a game has an outcome that is a unique NE, as in the case of joint confession in the PD, that must be its unique solution.
We can specify one class of games in which NE is always not only necessary but sufficient as a solution concept. These are finite perfect-information games that are also zero-sum. A zero-sum game in the case of a game involving just two players is one in which one player can only be made better off by making the other player worse off. Tic-tac-toe is a simple example of such a game: any move that brings one player closer to winning brings her opponent closer to losing, and vice-versa.
In tic-tac-toe, this is a draw. For one thing, it is highly unlikely that theorists have yet discovered all of the possible problems. However, we can try to generalize the issues a bit. Consider the strategic-form game below taken from Kreps , p.
This game has two NE: s1-t1 and s2-t2. Note that no rows or columns are strictly dominated here. But if Player I is playing s1 then Player II can do no better than t1, and vice-versa; and similarly for the s2-t2 pair. If NE is our only solution concept, then we shall be forced to say that either of these outcomes is equally persuasive as a solution.
Note that this is not like the situation in the PD, where the socially superior situation is unachievable because it is not a NE. In the case of the game above, both players have every reason to try to converge on the NE in which they are better off. Consider another example from Kreps , p. Here, no strategy strictly dominates another. So should not the players and the analyst delete the weakly dominated row s2?
When they do so, column t1 is then strictly dominated, and the NE s1-t2 is selected as the unique solution. However, as Kreps goes on to show using this example, the idea that weakly dominated strategies should be deleted just like strict ones has odd consequences.
Suppose we change the payoffs of the game just a bit, as follows:. Note that this game, again, does not replicate the logic of the PD. This is not true of s2-t1 in the present game. If the possibility of departures from reliable economic rationality is taken seriously, then we have an argument for eliminating weakly dominated strategies: Player I thereby insures herself against her worst outcome, s2-t2.
Of course, she pays a cost for this insurance, reducing her expected payoff from 10 to 5. In the case just discussed, elimination of weakly dominated strategies is one possible refinement, since it refines away the NE s2-t1, and correlation is another, since it refines away the other NE, s1-t2, instead.
In principle, there seems to be no limit on the number of refinements that could be considered, since there may also be no limits on the set of philosophical intuitions about what principles a rational agent might or might not see fit to follow or to fear or hope that other players are following. We now digress briefly to make a point about terminology. This reflects the fact the revealed preference approaches equate choices with economically consistent actions, rather than being intended to refer to mental constructs.
Historically, there was a relationship of comfortable alignment, though not direct theoretical co-construction, between revealed preference in economics and the methodological and ontological behaviorism that dominated scientific psychology during the middle decades of the twentieth century.
Applications also typically incorporate special assumptions about utility functions, also derived from experiments. For example, players may be taken to be willing to make trade-offs between the magnitudes of their own payoffs and inequalities in the distribution of payoffs among the players.
We will turn to some discussion of behavioral game theory in Section 8. For the moment, note that this use of game theory crucially rests on assumptions about psychological representations of value thought to be common among people. For a proposed new set of conventions to reduce this labeling chaos, see Ross , pp. Non-psychological game theorists tend to take a dim view of much of the refinement program.
This is for the obvious reason that it relies on intuitions about which kinds of inferences people should find sensible. Behavioral game theory, by contrast, can be understood as a refinement of game theory, though not necessarily of its solution concepts, in a different sense.
It motivates this restriction by reference to inferences, along with preferences, that people do find natural , regardless of whether these seem rational , which they frequently do not. Non-psychological and behavioral game theory have in common that neither is intended to be normative—though both are often used to try to describe norms that prevail in groups of players, as well to explain why norms might persist in groups of players even when they appear to be less than fully rational to philosophical intuitions.
Let us therefore group non-psychological and behavioral game theorists together, just for purposes of contrast with normative game theorists, as descriptive game theorists. Descriptive game theorists are often inclined to doubt that the goal of seeking a general theory of rationality makes sense as a project. Institutions and evolutionary processes build many environments, and what counts as rational procedure in one environment may not be favoured in another.
On the other hand, an entity that does not at least stochastically i. To such entities game theory has no application in the first place. This does not imply that non-psychological game theorists abjure all principled ways of restricting sets of NE to subsets based on their relative probabilities of arising.
In particular, non-psychological game theorists tend to be sympathetic to approaches that shift emphasis from rationality onto considerations of the informational dynamics of games.
We should perhaps not be surprised that NE analysis alone often fails to tell us much of applied, empirical interest about strategic-form games e. Since each player chooses between two actions at each of two information sets here, each player has four strategies in total. The first letter in each strategy designation tells each player what to do if he or she reaches their first information set, the second what to do if their second information set is reached. If you examine the matrix in Figure 10, you will discover that LL, RL is among the NE.
This is a bit puzzling, since if Player I reaches her second information set 7 in the extensive-form game, she would hardly wish to play L there; she earns a higher payoff by playing R at node 7.
In analyzing extensive-form games, however, we should care what happens off the path of play, because consideration of this is crucial to what happens on the path. We are throwing away information relevant to game solutions if we ignore off-path outcomes, as mere NE analysis does. Notice that this reason for doubting that NE is a wholly satisfactory equilibrium concept in itself has nothing to do with intuitions about rationality, as in the case of the refinement concepts discussed in Section 2.
Begin, again, with the last subgame, that descending from node 7. He chooses L. At node 5 II chooses R. Note that, as in the PD, an outcome appears at a terminal node— 4, 5 from node 7—that is Pareto superior to the NE. Again, however, the dynamics of the game prevent it from being reached. It gives an outcome that yields a NE not just in the whole game but in every subgame as well. This is a persuasive solution concept because, again unlike the refinements of Section 2.
It does, however, assume that players not only know everything strategically relevant to their situation but also use all of that information. An agent playing a subgame perfect strategy simply chooses, at every node she reaches, the path that brings her the highest payoff in the subgame emanating from that node.
A main value of analyzing extensive-form games for SPE is that this can help us to locate structural barriers to social optimization. In our current example, Player I would be better off, and Player II no worse off, at the left-hand node emanating from node 7 than at the SPE outcome. The main techniques are reviewed in Hurwicz and Reiter , the first author of which was awarded the Nobel Prize for his pioneering work in the area.
Many readers, but especially philosophers, might wonder why, in the case of the example taken up in the previous section, mechanism design should be necessary unless players are morbidly selfish sociopaths. We have seen that in the unique NE of the PD, both players get less utility than they could have through mutual cooperation.
This may strike you, even if you are not a Kantian as it has struck many commentators as perverse. To begin with they have no regard for the social good, and then they shoot themselves in the feet by being too untrustworthy to respect agreements. This way of thinking is very common in popular discussions, and badly mixed up. To dispel its influence, let us first introduce some terminology for talking about outcomes.
Welfare economists typically measure social good in terms of Pareto efficiency. Now, the outcome 3,3 that represents mutual cooperation in our model of the PD is clearly Pareto superior to mutual defection; at 3,3 both players are better off than at 2,2. So it is true that PDs lead to inefficient outcomes. This was true of our example in Section 2.
However, inefficiency should not be associated with immorality. A utility function for a player is supposed to represent everything that player cares about , which may be anything at all.
As we have described the situation of our prisoners they do indeed care only about their own relative prison sentences, but there is nothing essential in this. What makes a game an instance of the PD is strictly and only its payoff structure. Thus we could have two Mother Theresa types here, both of whom care little for themselves and wish only to feed starving children.
But suppose the original Mother Theresa wishes to feed the children of Calcutta while Mother Juanita wishes to feed the children of Bogota. Our saints are in a PD here, though hardly selfish or unconcerned with the social good. In that case, this must be reflected in their utility functions, and hence in their payoffs. But all this shows is that not every possible situation is a PD; it does not show that selfishness is among the assumptions of game theory.
Agents who wish to avoid inefficient outcomes are best advised to prevent certain games from arising; the defender of the possibility of Kantian rationality is really proposing that they try to dig themselves out of such games by turning themselves into different kinds of agents. In general, then, a game is partly defined by the payoffs assigned to the players. In any application, such assignments should be based on sound empirical evidence. Our last point above opens the way to a philosophical puzzle, one of several that still preoccupy those concerned with the logical foundations of game theory.
It can be raised with respect to any number of examples, but we will borrow an elegant one from C. Bicchieri Consider the following game:. To see this, backward induct again. At node 10, I would play L for a payoff of 3, giving II a payoff of 1. II can do better than this by playing L at node 9, giving I a payoff of 0. I can do better than this by playing L at node 8; so that is what I does, and the game terminates without II getting to move. Player I plays L at node 8 because she knows that Player II is economically rational, and so would, at node 9, play L because Player II knows that Player I is economically rational and so would, at node 10, play L.
Both players use backward induction to solve the game; backward induction requires that Player I know that Player II knows that Player I is economically rational; but Player II can solve the game only by using a backward induction argument that takes as a premise the failure of Player I to behave in accordance with economic rationality.
This is the paradox of backward induction. That is, a player might intend to take an action but then slip up in the execution and send the game down some other path instead. In our example, Player II could reason about what to do at node 9 conditional on the assumption that Player I chose L at node 8 but then slipped.
Gintis a points out that the apparent paradox does not arise merely from our supposing that both players are economically rational. It rests crucially on the additional premise that each player must know, and reasons on the basis of knowing, that the other player is economically rational. A player has reason to consider out-of-equilibrium possibilities if she either believes that her opponent is economically rational but his hand may tremble or she attaches some nonzero probability to the possibility that he is not economically rational or she attaches some doubt to her conjecture about his utility function.
We will return to this issue in Section 7 below. The paradox of backward induction, like the puzzles raised by equilibrium refinement, is mainly a problem for those who view game theory as contributing to a normative theory of rationality specifically, as contributing to that larger theory the theory of strategic rationality.
This involves appeal to the empirical fact that actual agents, including people, must learn the equilibrium strategies of games they play, at least whenever the games are at all complicated.
Instead of constructing utility functions on the basis of single episodes, we must do so on the basis of observed runs of behavior once it has stabilized , signifying maturity of learning for the subjects in question and the game in question.
As a result, when set into what is intended to be a one-shot PD in the experimental laboratory, people tend to initially play as if the game were a single round of a repeated PD. The repeated PD has many Nash equilibria that involve cooperation rather than defection.
Thus experimental subjects tend to cooperate at first in these circumstances, but learn after some number of rounds to defect. The experimenter cannot infer that she has successfully induced a one-shot PD with her experimental setup until she sees this behavior stabilize.
If players of games realize that other players may need to learn game structures and equilibria from experience, this gives them reason to take account of what happens off the equilibrium paths of extensive-form games. Of course, if a player fears that other players have not learned equilibrium, this may well remove her incentive to play an equilibrium strategy herself. This raises a set of deep problems about social learning Fudenberg and Levine The crucial answer in the case of applications of game theory to interactions among people is that young people are socialized by growing up in networks of institutions , including cultural norms.
Novices must then only copy those whose play appears to be expected and understood by others. As noted in Section 2. Given the complexity of many of the situations that social scientists study, we should not be surprised that mis-specification of models happens frequently. Applied game theorists must do lots of learning, just like their subjects. The paradox of backward induction is one of a family of paradoxes that arise if one builds possession and use of literally complete information into a concept of rationality.
Consider, by analogy, the stock market paradox that arises if we suppose that economically rational investment incorporates literally rational expectations: assume that no individual investor can beat the market in the long run because the market always knows everything the investor knows; then no one has incentive to gather knowledge about asset values; then no one will ever gather any such information and so from the assumption that the market knows everything it follows that the market cannot know anything!
The dictator and ultimatum games hold important lessons for issues such as charitable giving and philanthropy.
The worst possible outcome is realized if nobody volunteers. For example, consider a company in which accounting fraud is rampant , though top management is unaware of it. Being labeled as a whistleblower may also have some repercussions down the line. The centipede game is an extensive-form game in game theory in which two players alternately get a chance to take the larger share of a slowly increasing money stash. It is arranged so that if a player passes the stash to their opponent who then takes the stash, the player receives a smaller amount than if they had taken the pot.
The centipede game concludes as soon as a player takes the stash, with that player getting the larger portion and the other player getting the smaller portion. The game has a pre-defined total number of rounds, which are known to each player in advance. Of course, we are social beings who do cooperate and do care about the welfare of others, often at our own expense.
Game theory cannot account for the fact that in some situations we may fall into a Nash equilibrium, and other times not, depending on the social context and who the players are. The "games" may involve how two competitor firms will react to price cuts by the other, whether a firm should acquire another, or how traders in a stock market may react to price changes. In theoretic terms, these games may be categorized as prisoner's dilemmas, the dictator game, the hawk-and-dove, and Bach or Stravinsky.
Like many economic models, game theory also contains a set of strict assumptions that must hold for the theory to make good predictions in practice. First, all players are utility-maximizing rational actors that have full information about the game, the rules, and the consequences. Players are not allowed to communicate or interact with one another. Possible outcomes are not only known in advance but also cannot be changed.
The Nash equilibrium provides the solution concept in a non-cooperative adversarial game. It is named after John Nash who received the Nobel in for his work. Princeton University Press. Proceedings of the National Academy of Sciences. Behavioral Economics.
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Economics Behavioral Economics. What Is Game Theory? Key Takeaways Game theory is a theoretical framework to conceive social situations among competing players and produce optimal decision-making of independent and competing actors in a strategic setting. Scenarios include the prisoner's dilemma and the dictator game among many others. What are the games being played in game theory? What are some of the assumptions about these games?
What is a Nash equilibrium? Who came up with game theory?
Game Theory Definition - investopedia.com
John Harsanyi: An economist who won the Nobel Memorial Prize in 1994 along with John Nash and Reinhard Selten for his research on game theory, a mathematical system for predicting the outcomes of ...
Game theory provides a general framework to describe and analyze how individuals behave in such “strategic” situations. This course focuses on the key concepts in game theory, and attempts to outline the informal basic ideas that are often hidden behind mathematical definitions. Game theory is the process of modeling the strategic interaction between two or more players in a situation containing set rules and outcomes. While used in a number of disciplines, game theory is. by movies such as "A Beautiful Mind," game theory is the mathematical modeling of strategic interaction among rational (and irrational) Beyond what we call `games' in common such as chess, poker, soccer, etc., it includes the modeling of conflict among nations, political campaigns, competition among firms, and.
What are some of the assumptions about these games?