Game Theory 101: The Rationality of War

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His argument was rejected on the grounds that it has a devastating implication: to empty epistemology of its normative character Bishop The objection was based on the belief that psychology is a strictly descriptive science. Parts of psychology are normative. Theoretical and empirical results are used to prescribe what means people should use to achieve goals.

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People ought to use natural frequencies to improve Bayesian reasoning. If conditions E hold, then people ought to rely on one-reason heuristics rather than on linear models. These examples demonstrate that psychology is both a descriptive and a normative discipline. Thus, naturalizing epistemology by taking psychology into account does not imply emptying epistemology of its normative content Bishop and Trout What it does do is question the divide of disciplines into strictly normative and descriptive ones, and the associated justification of a priori, intuitive, and categorical norms Schurz Tests of predictive power, such as in out-of-sample prediction, as opposed to data fitting.

Epistemic rationality is often seen as promoting coherent beliefs rather than incoherent ones, and true beliefs rather than false beliefs. Implicitly, the assumption is often that coherence is associated with truth. Yet, as we have seen, there is lack of evidence both that coherent beliefs are correlated with true beliefs and that incoherence has substantial costs.

Furthermore, in situations of uncertainty, coherence is no longer clearly defined. All of this requires extending epistemic goals from coherence to other goals such as the accuracy of judgment truth , its speed how fast a judgment can be made , and frugality how many cues need to be searched before a judgment can be made. To decide about the ecological rationality of a heuristic, these goals need to be clearly defined.

For instance, accuracy can be decomposed into hit rate such as the proportion of patients who are diagnosed as having a disease among those who actually have the disease and false alarm rate the proportion of patients who are diagnosed as having a disease among those who do not have it. Heuristics can be designed to achieve a desired balance between the two rates. For instance, the question of which fast-and-frugal trees are ecologically rational for which balance of hits and misses has been solved Luan et al. According to one view, heuristics are subject to an accuracy — effort trade - off Kahneman ; Shah and Oppenheimer : Heuristics save effort but at the cost of accuracy.

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In this view, the rationality behind relying on heuristics lies solely in reducing effort, winning time, and avoiding information search. This is probably the dominant interpretation of heuristics in philosophy and the social sciences. Accuracy — effort trade - offs hold in situations of risk where the optimal course of action is calculable but not necessarily in situations of uncertainty. Under uncertainty, less - is - more effects exist.

Game Theory 101: Rationality

Less - is - more effect Assume two strategies, P and T. P uses only a proper subset of the information that T uses. If P makes more accurate predictions, this is called a less-is-more effect. Less-is-more does not imply a monotonic relationship between decreasing effort and increasing accuracy. Rather, there is generally an inverse U-shaped relationship between effort and accuracy, meaning that after some point on this curve, more information search is not only costly but also reduces accuracy.

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This phenomenon has been discussed as apparent epistemic irresponsibility Bishop Less-is-more effects should not occur if accuracy—effort trade-offs were generally true. To understand why and when less-is-more effects can be expected, I will first introduce the bias—variance dilemma as a general alternative to the accuracy—effort trade-off in situations of uncertainty, and thereafter a specific analysis of the ecological rationality for the take-the-best heuristic and similar one-good-reason heuristics.

A visual analogy of the two components of prediction error: bias and variance.

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The example illustrates situations where biased minds can make more accurate predictions than unbiased ones do Gigerenzer The darts on the left-hand dartboard show a systematic bias but low variance. A moderate bias with low variance left may lead to better accuracy than would a zero bias with high variance. The variance component of the error corresponds to the concept of overfitting. A model with many free parameters may fit the data perfectly but predict worse than simpler models Forster and Sober Heuristics can reduce error due to variance in several ways.

A total reduction of variance can be achieved by a heuristic that is insensitive to data by using no adjustable parameters. In Fig. A hiatus heuristic with a fixed hiatus is an example. Such a heuristic will avoid being overly sensitive to the peculiarities of the sample information available. In other situations, however, ignoring the evidence from samples may increase bias to the extent that the total error increases. Heuristics that learn from samples can still reduce error due to variance by 1 ignoring valid predictors, 2 not estimating weights, and 3 not estimating the covariance matrix between cues or reasons and treating the cues as independent.

Heuristics such as take-the-best combine all three of these principles. The purpose of the following case study is to explain in more detail the study of ecological rationality and how it differs from axiomatic rationality. Consider the task of inferring which of two alternatives has the larger value on some criterion, such as which contestant will win a tennis match or which high school will have more drop-outs.

To make this inference, there are n cues or reasons. Experiments showed that people tend not to use all cues, even if each is valid, and often proceed in a lexicographic order. The term lexicographic has its origin in the way one looks up a word in a lexicon; first, one searches for the first letter, then the second, and so on. In decision theory, the term refers to a process in which one looks up cues in sequential order and can stop search immediately after the first or a later cue if a stopping rule is satisfied.

Because lexicographic choice cannot be mapped into a utility function, and may not conform to the choice axioms, it has been interpreted a priori as irrational. Search rule Search cues in order of their validity v. Stopping rule Stop search on finding the first cue that discriminates between the alternatives i. Decision rule Infer that the alternative with the positive cue value 1 has the higher criterion value. Numerous studies have shown that in situations where take-the-best is ecologically rational see below , a large proportion of people tend to rely on it.

This includes student populations e. However, people who rely on this heuristic appear to commit several transgressions against logical rationality: First, as mentioned before, lexicographic choices cannot be represented by a utility function. Second, choices appear to violate the principle of total evidence by 1 ignoring all dependencies between cues, that is, the entire covariance matrix, when ordering the cues by v , and 2 ignoring all other cues after the first cue is found that allows for a judgment, leaving valid information on the table.

Each of these properties has been interpreted as a cognitive error. However, Keeney and Raiffa argued solely from their a priori normative view of axiomatic rationality, without any testing. Take-the-best corresponds to the left-hand dart board in Fig. The amount of bias take-the-best has depends on the structure of the environment. What are the environmental conditions E that take-the-best and similar one-reason heuristics can exploit? The term environment refers to the alternatives, cues, criteria, and other factors relevant for the decision maker.

The environment determines the bias of a heuristic and other strategies. Because the true value may not be accessible and will differ from problem to problem, the question asked is a comparative one: Can we identify general environmental structures in which the bias of take-the-best is equal to that of linear models? Answering this question helps identify situations where take-the-best can be expected to be more accurate than linear models, that is, when the bias is the same but the heuristic generates less error by variance, resulting in smaller total error Eq.

Consider a choice between objects A and B , based on n cues, where the value of the i th cue is represented by x i and weighted in the linear payoff function by w i. To simplify, assume that the cues are binary and the weights are nonnegative. We know of three environmental conditions where the bias of take-the-best is the same as that of linear models: noncompensatoriness, dominance, and cumulative dominance Gigerenzer These features can be seen as ecological axioms:.

If the weights are noncompensatory, then a linear rule with the same order of cues will always lead to the same choice as a lexicographic rule Martignon and Hoffrage Take the example above. Thus, if noncompensatoriness holds, then a lexicographic heuristic will have the same bias as any linear model with the same order of cues. Dominance If alternative A has a value higher than or equal to alternative B on all n cues and a higher value on at least one cue, then alternative A dominates alternative B.

Thus, if dominance holds, then a lexicographic heuristic will have the same bias as any linear model. Cumulative Dominance The cumulative profile of an alternative consists of n values, where the i th value is the sum of the first i values. Alternative A cumulatively dominates B if its cumulative profile exceeds or equals the cumulative profile of B in every term and exceeds it in at least one term Baucells et al.

Week 2: Mixed-Strategy Nash Equilibrium

Dominance implies cumulative dominance, but not vice versa. If cumulative dominance holds, then a linear rule with the same order of cues predicts the cumulative dominant object, just as a lexicographic rule does. In that case, a lexicographic heuristic can be said to be ecologically rational relative to any linear model because one can expect at least the same accuracy with less effort.

The three conditions explain when to expect this.

The Rationality of War

They do not explain when and why heuristics can predict more accurately, which can, however, be understood from Eq. If conditions E hold—noncompensatoriness, dominance, or cumulative dominance—then people ought to rely on take-the-best rather than on linear models. The data sets spanned areas as diverse as biology, business, computer science, ecology, economics, education, engineering, and medicine, among others.

The number of cues ranged from 3 to 21, which were numeric or binary; the number of objects alternatives ranged from 12 to , which resulted in a number of possible pairwise comparisons ranging from 66 to , The result was surprising. This means that in the majority of the cases, the lexicographic heuristics had the same bias as a linear model. Together with their potential for reducing variance, these results explain when and why simple heuristics can outperform linear models in prediction.

My first conjecture is that outside of these stable, well-defined situations, axiomatic rationality has no normative force. My second, related conjecture is that despite the widespread interpretations of violations of logical rationality as signs of human irrationality, there exists little to no empirical evidence that these violations would incur substantial costs such as diminished health, wealth, or happiness.

The third conjecture is that psychology can inform prescriptions for what people should do to achieve a given goal, which is the topic of the theory of ecological rationality. The study of heuristics extends axiomatic rationality from small worlds to situations of uncertainty and intractability, and from coherence to performance goals such as predictive accuracy, speed, and frugality.

EconoSpeak: Game theory and rationality -- the road less traveled

Developing such a theory entails the descriptive study of how individuals and institutions actually make decisions and the prescriptive study of the ecological rationality of heuristics. It also teaches us that in situations of uncertainty, less information can be more beneficial. These results conflict with the ideal of an a priori, categorical interpretation of axiomatic rationality and, more generally, logical rationality. The very existence of uncertainty and intractability defies choice axioms or logical rules as universal norms.

Ecological rationality, in contrast, emphasizes the adaptive character of rules or heuristics to reach goals.

Game Theory 101: The Rationality of War Game Theory 101: The Rationality of War
Game Theory 101: The Rationality of War Game Theory 101: The Rationality of War
Game Theory 101: The Rationality of War Game Theory 101: The Rationality of War
Game Theory 101: The Rationality of War Game Theory 101: The Rationality of War
Game Theory 101: The Rationality of War Game Theory 101: The Rationality of War
Game Theory 101: The Rationality of War Game Theory 101: The Rationality of War
Game Theory 101: The Rationality of War Game Theory 101: The Rationality of War
Game Theory 101: The Rationality of War Game Theory 101: The Rationality of War
Game Theory 101: The Rationality of War Game Theory 101: The Rationality of War

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