Values for Strategic Games in which Players Cooperate

Luisa CarpenteBalbina Casas

Ignacio García JuradoAnne van den Nouweland

Born 28 December 1903, Budapest, Hungary;

Died 8 February 1957, Washington DC

John von Neumann

J. von Neumann and O. Morgenstern (1944) “Theory of Games and Economic Behavior”. Princeton University

Press.

“...it is apparent from the evidence presented above that all the technical aspects of the theory may be credited to von Neumann.”

“Morgenstern’s role was crucial. (…) He focused the latter (von Neumann) attention, he acted as a spark.”

Leonard RJ (1995) From Parlor Games to Social Science: von Neumann, Morgenstern and the Creation of Game Theory. Journal of Economic Literature 33, 730-761.

P I

P -1 1

I 1 -1

Matching Pennies

P P L

P -1 1 0

I 1 -1 0

L 0 0 0

The Minimax Theorem

Theorem (von Neumann, 1928)

The von Neumann and Morgenstern Procedure

A strategic game:

For every non-empty coalition S different from N:

The TU-game associated with g:

• 1944. von Neumann and Morgenstern introduce the stable sets as a solution concept for TU-games.

• 1950. John Nash proposes the non-cooperative analysis of strategic games. His paper on equilibrium points is strongly influential in modern economic theory.

• 1953. Lloyd Shapley provides the Shapley value as a solution concept for TU-games. This concept has become central in cooperative game theory.

There may be games which themselves -by virtue of the rules of the game (...)- provide the mechanism for agreements and their enforcement. But we cannot base our considerations on this possibility since a game need not provide this mechanism; (...) Thus there seems no escape from the necessity of considering agreements concluded outside the game.

J. von Neumann and O. Morgenstern in “Theory of Games and Economic Behavior”.

The terminology that is used sometimes gives rise to confusion; it is not the case that in non-cooperative games players do not wish to cooperate and that in cooperative games players automatically do so. The difference instead is in the level of detail of the model; non-cooperative models assume that all the possibilities for cooperation have been included as formal moves in the game, while cooperative models are ''incomplete'' and allow players to act outside of the detailed rules that have been specified.

E. van Damme and D. Furth

“Game theory and the market”. In: P. Borm and H. Peters (eds.) Chapters in Game Theory. Kluwer Academic Publishers. 2002.

Other procedures in...

• Harsanyi JC (1963). A simplified bargaining model for the n-person cooperative game. International Economic Review 4:58-71.

• Myerson RB (1991). Game Theory, Analysis of Conflict. Harvard University Press.

• Bergantiños G and García-Jurado I (1995). A comparative study of several characteristic functions associated with a normal form game (in Spanish). Investigaciones Económicas 19:127-138.

• ...

A new procedure for settings in which mixed strategies are

not possible or reasonableA strategic game:

For every non-empty coalition S different from N:

The TU-game associated with g:

• We would like to provide axiomatic foundations for the von Neumann and Morgenstern’s procedure and for our new procedure, both based on value ideas.

• Inspiration is taken from:– Vilkas EI (1963). Axiomatic definition of the

value of a matrix game. Theory of Probability and its Applications 8:304-307.

– Tijs SH (1981). A characterization of the value of zero-sum two-person games. Naval Research Logistics Quarterly 28:153-156.

An evaluation function

Theorem (Vilkas (1963)). The value function V is the unique evaluation function that satisfies objectivity (A1), monotonicity (A2), row dominance (A3) and symmetry (A4).

Theorem. The value function V is the unique evaluation function that satisfies objectivity (A1), monotonicity (A2), row dominance (A3) and column dominance (A5).

There are other characterizations of the value function

Theorem. The value function V is the unique evaluation function that satisfies objectivity (A1), row dominance (A3), column dominance (A5), row elimination (A6) and column elimination (A7).

Norde H and Voorneveld M (2003). Axiomatizations of the value of matrix games. CentER Discussion Paper 2003-17. Tilburg University.

Hart S, Modica S and Schmeidler D (1994). A Neo Bayesian Foundation of the Maxmin Value for Two-Person Zero-Sum Games. International Journal of Game Theory 23, 347-358.

Theorem. The value function V is the unique evaluation function that satisfies objectivity (A1), monotonicity (A2), row dominance (A3) and column dominance (A5).

Is there an analogous characterization of the lower value function?

5 0

1 43 2

A V(A)=2

A’5 0

1 4 V(A’)=1

V does not satisfy row dominance.

Some properties satisfied by the lower value function

The lower value function also satisfies objectivity (A1), monotonicity (A2) and column dominance (A5).

1 0 3

1 3 0A A’ 0 3

3 0

The value function does not satisfy A9 because V(A)=1 and V(A’)=3/2.

Theorem. The lower value function V is the unique evaluation function that satisfies objectivity (A1), monotonicity (A2), weak row dominance (A8) and strong column dominance (A9).

Theorem. The lower value function V is the unique evaluation function that satisfies objectivity (A1), row elimination (A6), column elimination (A7), weak row dominance (A8) and strong column dominance (A9).

A procedure

Some open problems

• A comparative study among the several procedures.

• Characterizations of other procedures.

• A characterization of the Shapley value and other solution concepts in this setup.

Some open problems

• A comparative study among the several procedures.

• Characterizations of other procedures.

• A characterization of the Shapley value and other solution concepts in this setup.

Values for Strategic Games in which Players Cooperate

Luisa CarpenteBalbina Casas

Ignacio García JuradoAnne van den Nouweland