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Updated 20 hrs ago. Kemp names 2 new regents as chancellor impasse lingers. 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. 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. 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. The interest of philosophers in game theory is more often motivated by this ambition than is that of the economist or other scientist. 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. Now, almost all theorists agree that avoidance of strictly dominated strategies is a minimum requirement of economic rationality.
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. However, most games do not have this property. 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. First, there is the problem that in most non-zero-sum games, there is more than one NE, but not all NE look equally plausible as the solutions upon which strategically alert players would hit.
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. There, it makes sense to eliminate the most attractive outcome, joint refusal to confess, because both players have incentives to unilaterally deviate from it, so it is not an NE. 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. On the other hand, we might imagine that the players could communicate before playing the game and agree to play correlated strategies so as to coordinate on s2-t1, thereby removing some, most or all of the uncertainty that encourages elimination of the weakly dominated row s1, and eliminating s1-t2 as a viable solution instead! Any proposed principle for solving games that may have the effect of eliminating one or more NE from consideration as solutions is referred to as a refinement of NE.
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. So which refinement is more appropriate as a solution concept?
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.
However, this usage is increasingly likely to cause confusion due to the more recent rise of behavioral game theory Camerer 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. We mean by this the kind of game theory used by most economists who are not revisionist behavioral economists. 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.
Like most scientists, non-psychological game theorists are suspicious of the force and basis of philosophical assumptions as guides to empirical and mathematical modeling.
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. Equilibrium selection issues are often more fruitfully addressed in the context of extensive-form games. In order to deepen our understanding of extensive-form games, we need an example with more interesting structure than the PD offers. This game is not intended to fit any preconceived situation; it is simply a mathematical object in search of an application.
If you are confused by this, remember that a strategy must tell a player what to do at every information set where that player has an action. 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. 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.
But, as noted earlier, it is best to be careful not to confuse the general normative idea of rationality with computational power and the possession of budgets, in time and energy, to make the most of it.
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. If our players wish to bring about the more socially efficient outcome 4,5 here, they must do so by redesigning their institutions so as to change the structure of the game.
The enterprise of changing institutional and informational structures so as to make efficient outcomes more likely in the games that agents that is, people, corporations, governments, etc. 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. This theme is explored with great liveliness and polemical force in Binmore , 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.
Surely, you may think, it simply results from a combination of selfishness and paranoia on the part of the players. 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:. The NE outcome here is at the single leftmost node descending from node 8. 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. A puzzle is then raised by Bicchieri along with other authors, including Binmore and Pettit and Sugden by way of the following reasoning.
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. What it means to say that people must learn equilibrium strategies is that we must be a bit more sophisticated than was indicated earlier in constructing utility functions from behavior in application of Revealed Preference Theory. 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.
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