Editing Shapley value (section)

== Aumann–Shapley value ==
In their 1974 book, [[Lloyd Shapley]] and [[Robert Aumann]] extended the concept of the Shapley value to infinite games (defined with respect to a [[atom (measure theory)|non-atomic]] [[measure (mathematics)|measure]]), creating the diagonal formula.<ref name=":0">{{cite book |first1=Robert J. |last1=Aumann |first2=Lloyd S. |last2=Shapley |title=Values of Non-Atomic Games |publisher=Princeton Univ. Press |location=Princeton |year=1974 |isbn=0-691-08103-4 }}</ref> This was later extended by [[Jean-François Mertens]] and [[Abraham Neyman]].

As seen above, the value of an n-person game associates with each player the expectation of their contribution to the worth of the coalition of players before them in a random ordering of all the players. When there are many players and each individual plays only a minor role, the set of all players preceding a given one is heuristically thought of as a good sample of all players. The value of a given infinitesimal player {{mvar|ds}} is then defined as "their" contribution to the worth of a "perfect" sample of all the players.

Symbolically, if {{mvar|v}} is the coalitional worth function that associates each coalition {{mvar|c}} with its value, and each coalition {{mvar|c}} is a measurable subset of the measurable set {{mvar|I}} of all players, that we assume to be <math>I=[0,1]</math> without loss of generality, the value <math>(Sv)(ds)</math> of an infinitesimal player {{mvar|ds}} in the game is

: <math>
(Sv)(ds) = \int_0^1 (\, v(tI + ds)- v(tI)\,)\,dt.
</math>

Here {{mvar|tI}} is a perfect sample of the all-player set {{mvar|I}} containing a proportion {{mvar|t}} of all the players, and <math>tI+ ds</math> is the coalition obtained after {{mvar|ds}} joins {{mvar|tI}}. This is the heuristic form of the diagonal formula.<ref name=":0" />

Assuming some regularity of the worth function, for example, assuming {{mvar|v}}  can be represented as differentiable function of a non-atomic measure on  {{mvar|I}}, {{mvar|&mu;}}, <math>v(c)=f(\mu(c))</math> with density function <math>\varphi</math>, with <math>\mu(c)=\int 1_c(u)\varphi(u)\,du,</math> where <math> 1_c(\bullet)</math> is the characteristic function of {{mvar|c}}. Under such conditions

: <math>\mu(tI)=t\mu(I)
</math>,

as can be shown by approximating the density by a step function and keeping the proportion {{mvar|t}} for each level of the density function, and

: <math>
v(tI + ds)=f(t\mu(I))+f'(t\mu(I))\mu(ds)
.</math>

The diagonal formula has then the form developed by Aumann and Shapley (1974)

: <math>
(Sv)(ds) = \int_0^1 f'_{t\mu(I)}(\mu(ds)) \, dt
</math>

Above {{mvar|&mu;}} can be vector valued (as long as the function is defined and differentiable on the range of {{mvar|&mu;}}, the above formula makes sense).

In the argument above if the measure contains atoms <math>\mu(tI)=t\mu(I)</math> is no longer true—this is why the diagonal formula mostly applies to non-atomic games.

Two approaches were deployed to extend this diagonal formula when the function {{mvar|f}} is no longer differentiable. Mertens goes back to the original formula and takes the derivative after the integral thereby benefiting from the smoothing effect. Neyman took a different approach. Going back to an elementary application of Mertens's approach from Mertens (1980):<ref>{{cite journal |last=Mertens |first=Jean-François |year=1980 |title=Values and Derivatives |journal=[[Mathematics of Operations Research]] |volume=5 |issue=4 |pages=523–552 |jstor=3689325 |doi=10.1287/moor.5.4.523}}</ref>

: <math>
(Sv)(ds) =  \lim_{\varepsilon \to 0, \varepsilon>0} \frac{1}{\varepsilon}\int_0^{1-\varepsilon} (f(t+\varepsilon \mu(ds))-f(t)) \, dt
</math>

This works for example for majority games—while the original diagonal formula cannot be used directly. How Mertens further extends this by identifying symmetries that the Shapley value should be invariant upon, and averaging over such symmetries to create further smoothing effect commuting averages with the derivative operation as above.<ref>{{cite journal |last=Mertens |first=Jean-François |year=1988 |title=The Shapley Value in the Non Differentiable Case |journal=International Journal of Game Theory |volume=17 |issue=1 |pages=1–65 |doi=10.1007/BF01240834 |s2cid=118017018 }}</ref> A survey for non atomic value is found in Neyman (2002)<ref>Neyman, A., 2002. Value of Games with infinitely many Players, "Handbook of Game Theory with Economic Applications," Handbook of Game Theory with Economic Applications, Elsevier, edition 1, volume 3, number 3, 00. R.J. Aumann & S. Hart (ed.).[http://ratio.huji.ac.il/dp/neyman/values.pdf]</ref>