Epistemic Game Theory

Epistemic game theory ( English epistemic game theory ), otherwise called interactive epistemology ( English interactive epistemology ), formalizes assumptions about the faiths and knowledge of players regarding the rationality , behavior of opponents, their own knowledge and beliefs. These assumptions underlie the various decision concepts — the rules that predict the behavior of players and therefore the outcome of the game. Assumptions are often described at an intuitive level, and epistemic analysis is necessary to rigorously justify the use or non-use of a particular concept. Epistemic analysis allows us to clarify the intuitive description of assumptions, identifying their imperfections and non-obvious consequences, generalize intuition and outline the boundaries of applicability of concepts. At the same time, the epistemic theory of games is not the only and comprehensive approach to the substantiation of solution concepts, since sometimes epistemic conditions are excessively strong.

An example of a multitude of elementary events can be the strategies of other participants that he does not observe. One of the central elements of the epistemic theory is the hierarchy of faiths , with the help of which the conditions of rationality and the general belief in rationality are formalized. The hierarchy of faiths is a countable set of faiths, namely: faith in relation to the strategies of other participants, faith in relation to their faiths, etc. One of the first formal ways to build an endless hierarchy was proposed by John Harsagni . He introduced a type structure that gives each of the participants a set of possible states (types). The type of player is determined in accordance with the well-known distribution, however, its implementation is a priori known only to the owner of the type, or is unknown to anyone. Type, in particular, maps the player to a belief system about strategies and types of opponents.

Content

1 Faith and knowledge
- 1.1 Semantic representation
2 notes
- 2.1 Comments
- 2.2 Sources
- 2.3 Literature
- 2.4 Conformity of terms

Faith and Knowledge

In the epistemic theory of games, there are two approaches to modeling faith and knowledge. The semantic approach is based on set theory ^[1] , the syntactic approach is based on modal logic .

Semantic Presentation

Suppose there are many states ^[comm. ^one] $\Omega$ ${\ displaystyle \ Omega}$ . Under the state means an exhaustive description of the relevant characteristics of the world. Subsets $\Omega$ ${\ displaystyle \ Omega}$ are called events and the set of all events is indicated $2^{\Omega }$ ${\ displaystyle 2 ^ {\ Omega}}$ . There is an individual whose information about the world is limited. To model this uncertainty, a capability operator is introduced. $P:\Omega \rightarrow 2^{\Omega }$ ${\ displaystyle P: \ Omega \ rightarrow 2 ^ {\ Omega}}$ , associating each state with a subset of states. Being able $\omega \in \Omega$ ${\ displaystyle \ omega \ in \ Omega}$ , the individual only knows that he is in a subset $P(\omega )\subseteq \Omega$ ${\ displaystyle P (\ omega) \ subseteq \ Omega}$ . Couple $(\Omega ,P)$ ${\ displaystyle (\ Omega, P)}$ called the scale of ver .

An individual knows about the occurrence of a specific event only in the case of $P\omega \subseteq E$ ${\ displaystyle P \ omega \ subseteq E}$ . Feature Operator $P$ ${\ displaystyle P}$ has two properties:

(P1)\qquad \omega \in P\omega

{\ displaystyle (P1) \ qquad \ omega \ in P \ omega}

(P2)\qquad P\omega \cap \omega '=\emptyset

{\ displaystyle (P2) \ qquad P \ omega \ cap \ omega '= \ emptyset}

or

P\omega =\omega '

{\ displaystyle P \ omega = \ omega '}

Whence it follows that the sets ${P\omega |\omega \in \Omega }$ ${\ displaystyle {P \ omega | \ omega \ in \ Omega}}$ is a partition $\Omega$ ${\ displaystyle \ Omega}$ . Using the opportunity operator, you can define the knowledge operator $KE={\omega |P\omega \subseteq E}$ ${\ displaystyle KE = {\ omega | P \ omega \ subseteq E}}$ . It has the following properties.

(K1)\qquad K\Omega =\Omega

{\ displaystyle (K1) \ qquad K \ Omega = \ Omega}

(K2)\qquad K(E\cap F)=KE\cap KF

{\ displaystyle (K2) \ qquad K (E \ cap F) = KE \ cap KF}

(K3)\qquad KE\subseteq E

{\ displaystyle (K3) \ qquad KE \ subseteq E}

(K4)\qquad KE=KKE

{\ displaystyle (K4) \ qquad KE = KKE}

(K5)\qquad \neg K\neg KE\subseteq E

{\ displaystyle (K5) \ qquad \ neg K \ neg KE \ subseteq E}

Notes

Comments

↑ States are also called possible worlds .

Sources

↑ Halpern, JY Why Bother With Syntax?

Literature

De Finetti, Bruno. Foresight: Its logical laws, its subjective sources, volume Breakthroughs in Statistics: Foundations and Basic Theory, pages 134 {174. Springer-Verlag, 1992.
Dekel, Eddie & Siniscalchi, Marciano. Epistemic game theory (forthcoming in the Handbook of Game Theory, vol. 4.).
Harsanyi JC Games with incomplete information played by \ Bayesian "players, I-III. Part I. The basic model. Management Science, pages 159 {182, 1967.
Perea, A. From classical to epistemic game theory. International Game Theory Review Vol. 16, No. 1 (2014).
Savage LJ The foundations of statistics. Dover Pubns, 1972.

Matching Terms

Russian term	English term
possible peace	possible world
ver operator	belief operator
opportunity operator	possibility correspondence
event	event
state	state
scale of ver	belief frame

[2] States are also called possible worlds .

[1] Halpern, JY Why Bother With Syntax?