Here are the basic baseball facts for kids about the game itself: ..

With the arrival of the Internet and real-time information and large databases, fantasy baseball has thousands of different formats, rules and styles.

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We know from experience that, in situations such as this, people donot usually stand and dither in circles forever. As we'll see later,there is a rational solution—that is, a best rationalaction—available to both players. However, until the 1940sneither philosophers nor economists knew how to find itmathematically. As a result, economists were forced to treatnon-parametric influences as if they were complications on parametricones. This is likely to strike the reader as odd, since, as ourexample of the bridge-crossing problem was meant to show,non-parametric features are often fundamental features ofdecision-making problems. Part of the explanation for game theory'srelatively late entry into the field lies in the problems with whicheconomists had historically been concerned. Classical economists, suchas Adam Smith and David Ricardo, were mainly interested in thequestion of how agents in very large markets—wholenations—could interact so as to bring about maximum monetarywealth for themselves. Smith's basic insight, that efficiency is bestmaximized by agents freely seeking mutually advantageous bargains, wasmathematically verified in the twentieth century. However, thedemonstration of this fact applies only in conditions of‘perfect competition,’ that is, when individuals or firmsface no costs of entry or exit into markets, when there are noeconomies of scale, and when no agents' actions have unintendedside-effects on other agents' well-being. Economists always recognizedthat this set of assumptions is purely an idealization for purposes ofanalysis, not a possible state of affairs anyone could try (or shouldwant to try) to attain. But until the mathematics of game theorymatured near the end of the 1970s, economists had to hope that themore closely a market approximates perfect competition, themore efficient it will be. No such hope, however, can bemathematically or logically justified in general; indeed, as a strictgeneralization the assumption was shown to be false as far back as the1950s.


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Baseball and Cricket are two well-known members of the 'bat and ball' family of games

Resolution of this debate between Gintis and Binmore fortunately neednot wait upon discoveries about the deep human evolutionary past thatwe may never have. The models make rival empirical predictions of sometestable phenomena. If Gintis is right then there are limits, imposedby the discontinuity in hominid evolution, on the extent to whichpeople can learn to be self-regarding. This is the main significanceof the controversy discussed above over Henrich et al.'sinterpretation of their field data. Binmore's model of socialequilibrium selection also depends, unlike Gintis's, on widespreaddispositions among people to inflict second-order punishment onmembers of society who fail to sanction violators of social norms. shows using a game theory model that this is implausible ifpunishment costs are significant. However, argues that the widespread assumption in the literature thatpunishment of norm-violation must be costly results from failure toadequately distinguish between models of the original evolution ofsociality, on the one hand, and models of the maintenance anddevelopment of norms and institutions once an initial set of them hasstabilized. Finally, Ross also points out that Binmore's objectivesare as much normative as descriptive: he aims to show egalitarians howto diagnose the errors in conservative rationalisations of the statusquo without calling for revolutions that put equilibrium pathstability (and, therefore, social welfare) at risk. It is a soundprinciple in constructing reform proposals that they should be‘knave-proof’ (as Hume put it), that is, should becompatible with less altruism than might prevail inpeople. Thus, despite the fact that the majority of researchersworking on game-theoretic foundations of social organization presentlyappear to side with Gintis and the other members of the Henrich etal. team, Binmore's alternative model has some strongconsiderations in its favor. Here, then, is another issue along thefrontier of game theory application awaiting resolution in the yearsto come.


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Any proposed principle for solving games that may have the effect ofeliminating one or more NE from consideration as solutions is referred to as arefinement of NE. In the case just discussed, elimination ofweakly dominated strategies is one possible refinement, since itrefines away the NE s2-t1, and correlation is another, since itrefines away the other NE, s1-t2, instead. So which refinement is moreappropriate as a solution concept? People who think of game theory asan explanatory and/or normative theory of strategic rationality havegenerated a substantial literature in which the merits and drawbacksof a large number of refinements are debated. In principle, thereseems to be no limit on the number of refinements that could beconsidered, since there may also be no limits on the set ofphilosophical intuitions about what principles a rational agent mightor might not see fit to follow or to fear or hope that other playersare following.

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Given the unresolved complex relationship between learning theory andgame theory, the reasoning above might seem to imply that game theorycan never be applied to situations involving human players that arenovel for them. Fortunately, however, we face no such impasse. In apair of influential papers in the mid-to-late 1990s, McKelvey andPalfrey (, )developed the solution concept of quantal responseequilibrium (QRE). QRE is not a refinement of NE, in the sense ofbeing a philosophically motivated effort to strengthen NE by referenceto normative standards of rationality. It is, rather, a method forcalculating the equilibrium properties of choices made by playerswhose conjectures about possible errors in the choices of otherplayers are uncertain. QRE is thus standard equipment in the toolkitof experimental economists who seek to estimate the distribution ofutility functions in populations of real people placed in situationsmodeled as games. QRE would not have been practically serviceable inthis way before the development of econometrics packages such as Stata(TM) allowed computation of QRE given adequately powerful observationrecords from interestingly complex games. QRE is rarely utilized bybehavioral economists, and is almost never used by psychologists, inanalyzing laboratory data. In consequence, many studies by researchersof these types make dramatic rhetorical points by‘discovering’ that real people often fail to converge onNE in experimental games. But NE, though it is a minimalist solutionconcept in one sense because it abstracts away from much informationalstructure, is simultaneously a demanding empirical expectation if itimposed categorically (that is, if players are expected to play as ifthey are all certain that all others are playing NEstrategies). Predicting play consistent with QRE is consistentwith—indeed, is motivated by—the view that NE captures thecore general concept of a strategic equilibrium. One way of framingthe philosophical relationship between NE and QRE is as follows. NEdefines a logical principle that is well adapted fordisciplining thought and for conceiving new strategies for genericmodeling of new classes of social phenomena. For purposes ofestimating real empirical data one needs to be able to defineequilibrium statistically. QRE represents one way of doingthis, consistently with the logic of NE.