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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.
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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.
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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.
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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. 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. Most complex games that people play are already in progress among people who were socialized before them—that is, have learned game structures and equilibria Ross a.
Novices must then only copy those whose play appears to be expected and understood by others. Institutions and norms are rich with reminders, including homilies and easily remembered rules of thumb, to help people remember what they are doing Clark 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!
As we will see in detail in various discussions below, most applications of game theory explicitly incorporate uncertainty and prospects for learning by players. The extensive-form games with SPE that we looked at above are really conceptual tools to help us prepare concepts for application to situations where complete and perfect information is unusual. We cannot avoid the paradox if we think, as some philosophers and normative game theorists do, that one of the conceptual tools we want to use game theory to sharpen is a fully general idea of rationality itself.
But this is not a concern entertained by economists and other scientists who put game theory to use in empirical modeling. In real cases, unless players have experienced play at equilibrium with one another in the past, even if they are all economically rational and all believe this about one another, we should predict that they will attach some positive probability to the conjecture that understanding of game structures among some players is imperfect. This then explains why people, even if they are economically rational agents, may often, or even usually, play as if they believe in trembling hands.
Learning of equilibria may take various forms for different agents and for games of differing levels of complexity and risk. Incorporating it into game-theoretic models of interactions thus introduces an extensive new set of technicalities. For the most fully developed general theory, the reader is referred to Fudenberg and Levine ; the same authors provide a non-technical overview of the issues in Fudenberg and Levine A first important distinction is between learning specific parameters between rounds of a repeated game see Section 4 with common players, and learning about general strategic expectations across different games.
The latter can include learning about players if the learner is updating expectations based on her models of types of players she recurrently encounters.
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A major difficulty for both players and modelers is that screening moves might be misinterpreted if players are also incentivized to make moves to signal information to one another see Section 4. Finally, the discussion so far has assumed that all possible learning in a game is about the structure of the game itself. It was said above that people might usually play as if they believe in trembling hands.
They must make and test conjectures about this from their social contexts. Sometimes, contexts are fixed by institutional rules. In other markets, she might know she is expect to haggle, and know the rules for that too.
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Given the unresolved complex relationship between learning theory and game theory, the reasoning above might seem to imply that game theory can never be applied to situations involving human players that are novel for them. Fortunately, however, we face no such impasse. In a pair of influential papers in the mid-to-late s, McKelvey and Palfrey , developed the solution concept of quantal response equilibrium QRE.
QRE is not a refinement of NE, in the sense of being a philosophically motivated effort to strengthen NE by reference to normative standards of rationality.
It is, rather, a method for calculating the equilibrium properties of choices made by players whose conjectures about possible errors in the choices of other players are uncertain. QRE is thus standard equipment in the toolkit of experimental economists who seek to estimate the distribution of utility functions in populations of real people placed in situations modeled as games.
QRE would not have been practically serviceable in this way before the development of econometrics packages such as Stata TM allowed computation of QRE given adequately powerful observation records from interestingly complex games. QRE is rarely utilized by behavioral economists, and is almost never used by psychologists, in analyzing laboratory data.
But NE, though it is a minimalist solution concept in one sense because it abstracts away from much informational structure, is simultaneously a demanding empirical expectation if it is imposed categorically that is, if players are expected to play as if they are all certain that all others are playing NE strategies. Predicting play consistent with QRE is consistent with—indeed, is motivated by—the view that NE captures the core general concept of a strategic equilibrium. NE defines a logical principle that is well adapted for disciplining thought and for conceiving new strategies for generic modeling of new classes of social phenomena.
For purposes of estimating real empirical data one needs to be able to define equilibrium statistically. QRE represents one way of doing this, consistently with the logic of NE. The idea is sufficiently rich that its depths remain an open domain of investigation by game theorists. We will see later that there is an alternative interpretation of mixing, not involving randomization at a particular information set; but we will start here from the coin-flipping interpretation and then build on it in Section 3.
Our river-crossing game from Section 1 exemplifies this. Symmetry of logical reasoning power on the part of the two players ensures that the fugitive can surprise the pursuer only if it is possible for him to surprise himself. Suppose that we ignore rocks and cobras for a moment, and imagine that the bridges are equally safe. He must then pre-commit himself to using whichever bridge is selected by this randomizing device.
This fixes the odds of his survival regardless of what the pursuer does; but since the pursuer has no reason to prefer any available pure or mixed strategy, and since in any case we are presuming her epistemic situation to be symmetrical to that of the fugitive, we may suppose that she will roll a three-sided die of her own.
Note that if one player is randomizing then the other does equally well on any mix of probabilities over bridges, so there are infinitely many combinations of best replies. However, each player should worry that anything other than a random strategy might be coordinated with some factor the other player can detect and exploit. Since any non-random strategy is exploitable by another non-random strategy, in a zero-sum game such as our example, only the vector of randomized strategies is a NE.
Now let us re-introduce the parametric factors, that is, the falling rocks at bridge 2 and the cobras at bridge 3. Suppose that Player 1, the fugitive, cares only about living or dying preferring life to death while the pursuer simply wishes to be able to report that the fugitive is dead, preferring this to having to report that he got away.