Encyclopedia Of General Knowledge By Jwt Pdf

[Marilu Mandez]

Inour example above for "Common Knowledge versus Information Needs a Citation," the Hungary example contains both historic dates and geographic features which are considered common knowledge, or things you could easily look up in an encyclopedia.

Common knowledge also includes basic knowledge that is known by those in a specific discipline or field of study. When writing for that specific field, information that is basic and foundational sometimes does not need to be cited. For example:

If information is not considered common knowledge, it needs to be cited. Information, work, ideas, or interpretations that are not your own need to be cited. There are some key categories of information that always need to be cited and are not considered common knowledge:

Most of the examples in this section are familiar in the commonknowledge literature, although some of the details and interpretationspresented here are new. Readers may want to ask themselves what, ifany, distinctive aspects of mutual and common knowledge reasoning eachexample illustrates.

In this reduced game, Fiona is certain to gain a strictly higherpayoff by choosing \(D^1\) than if she chooses \(C^1\), so \(D^1\) isher unique best choice. Of course, when Fiona chooses \(D^1\), Alan,being rational, responds by choosing \(D^2\). If Fiona and Alan know:(i) that they are both rational, (ii) that they both know the payoffstructure of the game, and (iii) that they both know (i) and (ii),then they both can predict what the other will do at every node of theFigure 1.1.a game, and conclude that they can rule out the\(D^1,C^2\)-branch of the Figure 1.1.b game and analyze just thereduced game of the following figure:

The mutual knowledge assumptions required to construct a backwardsinduction solution to a game become more complex as the number ofstages in the game increases. To see this, consider the sequentialCentipede game depicted in the following figure:

To illustrate the idea of possibility sets, let us return to theBarbecue Problem described in Example 1.2. Suppose there are threediners: Cathy, Jennifer and Mark. Then there are 8 relevant states ofthe world, summarized by Table 2.1:

Note that \((\mathbfK^m_N(E))_m\ge 1\) is a decreasing sequence ofevents, in the sense that \(\mathbfK^m+1_N (E) \subseteq\mathbfK^m_N(E)\), for all \(m \ge 1\). It is also easy to checkthat if everyone knows \(E\), then \(E\) must be true, that is,\(\mathbfK^1_N (E) \subseteq E\). If \(\Omega\) is assumed to befinite, then if \(E\) is common knowledge at \(\omega\), this impliesthat there must be a finite \(m\) such that

If \(E = \mathbfK^1_N (E)\), then \(E\) is a public event(Milgrom 1981) or a common truism (Binmore and Brandenburger1989). Clearly, a common truism is common knowledge whenever itoccurs, since in this case \(E = \mathbfK^1_N (E) = \mathbfK^2_N(E) =\ldots\) , so \(E = \mathbfK^*_N (E)\). The proof ofProposition 2.17 shows that the common truisms are precisely theelements of \(\mathcalM\) and unions of elements of \(\mathcalM\),so any commonly known event is the consequence of a common truism.

So we have established that \(\mathbfK^*_N (E)\) is a fixed pointof the function \(f_E\) defined by \(f_E (X) = \mathbfK^1_N (E\cap X). f_E\) has other fixed points. For instance, any contradiction\(B \cap B^c = \varnothing\) is a fixed point of \(f_E\).[15] Note also that if \(A \subseteq B\), then \(E \cap A \subseteq E \capB\) and so

Proposition 2.18

Let \(C^*_N\) be the greatest fixed point of \(f_E.\) Then \(C^*_N(E) = K^*_N (E).\) (In Barwise (1988, 1989), \(E\) is definedto be common knowledge at \(\omega\) iff \(\omega \in C^*_N(E).)\)

Proof.

\(\mathbfG_N^*(E)\) denotes the proposition defined by \(G_1^*\) and\(G_2^*\) for a set \(N\) of \(A^*\)-symmetric reasoners, so we cansay that \(E\) is Lewis-common knowledge for the agents of \(N\) iff\(\omega \in \mathbfG_N^*(E)\).

Readers primarily interested in philosophical applications of commonknowledge may want to focus on the No Disagreement Theorem andConvention subsections. Readers interested in applications of commonknowledge in game theory may continue with the Strategic Form Games,and Games of Perfect Information subsections.

Suppose next that both agents are Bayesian rational, and that part ofwhat each agent knows is the payoff structure of the Intersectiongame. If the agents expect each other to follow \((s_2, s_2)\) andthey consequently coordinate successfully, are they then following aconvention? Not necessarily, contends Lewis in a subtle argument on p.59 of Convention. For while each agent knows the game andthat she is rational, still she might not attribute the same knowledgeto the other agent. If each agent believes that the other agent willfollow her end of the \((s_2, s_2)\) equilibrium mindlessly, then herbest response is to follow her end of \((s_2, s_2)\). But in this casethe agents coordinated as the result of their each falsely believingthat the other acts like an automaton, and Lewis thinks that anyproper account of convention must require that agents havecorrect beliefs about one another. In particular, Lewisrequires that each agent involved in a convention must have mutualexpectations that each is acting with the aim of coordinating with theother. The argument can be carried further on. What if both agentsbelieve that they will follow \((s_2, s_2)\), and believe that eachother will do so thinking that the other will choose \(s_2\)rationally and not mindlessly? Then, say, Liz would coordinate as theresult of her false second-order belief that Robert believes that Lizacts mindlessly. Similarly for third-order beliefs and so on for anyhigher order of knowledge.

Lewis includes the requirement that there be an alternate coordinationequilibrium \(R'\) besides the equilibrium \(R\) that all follow inorder to capture the fundamental intuition that how the agents whofollow a convention behave depends crucially upon how they expect theothers to behave.

Lewis formulated the notion of common knowledge as part of his generalaccount of conventions. In the years following the publication ofConvention, game theorists have recognized that anyexplanation of a particular pattern of play in a game dependscrucially on mutual and common knowledge assumptions. Morespecifically, solution concepts in game theory are bothmotivated and justified in large part by the mutual or commonknowledge the agents in the game have regarding their situation.

if Joanna and Lizzi have common knowledge of all of the payoffs atevery strategy combination, and they have common knowledge that bothare Bayesian rational, then any of the four pure strategy profiles isrationalizable. For if their beliefs about each other are defined bythe probabilities

If \(\boldsymbol\mu^*\) is an endogenous correlated equilibrium apure strategy combination \(\boldsymbols^* = (s_1^*, \ldots ,s_n^*)\in S\) is an endogenous correlated equilibrium strategycombination given \(\boldsymbol\mu^*\) if, and only if, foreach agent \(k \in N,\)

If \(\boldsymbol\mu^*\) is a strict equilibrium, then one canpredict which pure strategy profile the agents in a game will followgiven common knowledge of the game, rationality and\(\boldsymbol\mu^*.\) But if \(\boldsymbol\mu^*\) is such thatseveral distinct pure strategy profiles satisfy (3.iv) with respect to\(\boldsymbol\mu^*\), then one can no longer predict with certaintywhat the agents will do. For instance, in the Chicken game of Figure3.1, the belief distributions defined by \(\alpha_1 = \alpha_2 = 2/3\)together are a Nash equilibrium-in-beliefs. Given common knowledge ofthis equilibrium, either pure strategy is a best reply for each agent,in the sense that either pure strategy maximizes expected utility.Indeed, if agents can also adopt randomized or mixedstrategies at which they follow one of several pure strategiesaccording to the outcome of a chance experiment, then any of theinfinitely mixed strategies an agent might adopt in Chicken is a bestreply given \(\boldsymbol\mu^*\).[25] So the endogenous correlated equilibrium concept does not determinethe exact outcome of a game in all cases, even if one assumesprobabilistic consistency and independence so that the equilibrium isa Nash equilibrium.

When Joanna moves she is at her information set \(I^22 = \C^1,D^1\,\) that is, she moves knowing that Lizzi might have choseneither \(C^1\) or \(D^1\), so this game is an extensive formrepresentation of the Chicken game of Figure 3.1.

In a game of perfect information, each information set consists of asingle node in the game tree, since by definition at each state theagent who is to move knows exactly how her predecessors have moved. InExample 1.4 it was noted that the method of backwards induction can beapplied to any game of perfect information.[30] The backwards induction solution is the unique Nash equilibrium of agame of perfect information. The following result gives sufficientconditions to justify backwards induction play in a game of perfectinformation:

Proposition 3.12 (Bicchieri 1993)

In an extensive form game of perfect information, the agents followthe backwards induction solution if the following conditions aresatisfied for each agent \(i\) at each information set \(I^ik\):

In complete networks (networks in which all playerscommunicate with everyone else, as within the triangle in the kitenetwork) the information partitions of the players coincide, and theyare the finest partitions of the set of possible worlds. Hence, ifplayers have sufficiently low thresholds, such fact is commonly knownand there is an equilibrium in which all players revolt.

In fact, skeptical doubt about the attainability of common knowledgeis certainly possible. A strong skeptical argument has been recentlyput forth by Lederman (2018b). Lederman builds an argument meant toundermine the possibility of deriving the common knowledge hierarchy,as done in 2, on the basis of a public event or, as Ledermancalls it, public information. The principle that Ledermantargets is what he calls ideal common knowledge (or belief),that is: If \(p\) is public information in a group \(G\) then \(p\) iscommon knowledge in \(G\), provided the agents in \(G\) are idealreasoners. The argument rests on the privacy and interpersonalincomparability of mental states among agents, and although it isoffered in terms of perceptual knowledge, its scope goes beyondperception to question the possibility of common knowledge toutcourt.

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