Method and system for using cooperative game theory to resolve statistical and other joint effects

• US 20040083078A1
• Filed: 10/14/2003
• Published: 04/29/2004
• Est. Priority Date: 04/06/2001
• Status: Active Grant

• Current Assignee: Barry E. Feldman
• Original Assignee: Barry E. Feldman
• Inventors: Feldman, Barry E.
• Granted Patent: US 6,961,678 B2
• Time in Patent Office: Days
• US Class Current: 702/185
• CPC Class Codes: G06F 17/18 for evaluating statistical data , e.g. average values, frequency distributions, probability functions, regression analysis forecasting specially adapted for a specific administrative, business or logistic context G06Q10/04

Abstract

A method and system for cooperative resolution of statistical and other joint effects. The method and system include constructing controlled allocation games. Allocation patterns in a control game are used to determine value allocations in an allocation game. The method and system also includes approximating value functions for large cooperative games. The method and system may be used to construct statistical cooperative games and use cooperative game theory to resolve statistical joint effects in a variety of situations. The methods may be applicable to other types of joint effects problems such as those found in engineering, finance and other disciplines.

Claims

1. A method for allocating value among players in a cooperative allocation game in order to resolve joint effects in an allocation problem, comprising:

identifying a control game and its players;

identifying an allocation game and its players;

establishing a control relationship between players or coalitions in the control game and players or coalitions in the allocation game;

selecting a set of coalitions in the control game;

determining a set of worths for the selected set coalitions in the control game;

evaluating one or more control functions using the determined set of worths to determine a set of values for the one or more control functions in the control game;

selecting a set of coalitions in the allocation game;

determining a set of worths for the selected set of coalitions in the allocation game; and

combining the determined set of values for the one or more control functions in the control game with the determined set of worths for the selected set of coalitions in the allocation game to determine allocations to players in the allocation game.

2. The method of claim 1 further comprising a computer readable medium having storing therein instructions for causing a processor to execute the steps of the method.

3. The method of claim 1 wherein the allocation game and control game include the same set of players.

4. The method of claim 1 wherein the established control relationship between players in the control game and the allocation game is a default identity relationship.

5. The method of claim 1 wherein:

one of the one or more control functions is a value function of the control game;

a weighted value is used as a value function of the allocation game, or

a player'"'"'s value in the control game is used as the player'"'"'s weight in the allocation game.

6. The method of claim 5 wherein the weighted value is a weighted Shapley value.

7. The method of claim 1 wherein:

the one or more control functions are ordered worth products of coalitions in the control game;

the set of worths of the selected coalitions in the allocation game are used to determine marginal contributions of individual players;

ordered worth products are used to weight players'"'"' marginal contributions in the allocation game, or

a value allocated to an individual player in the allocation game is determined by the weighted marginal contributions.

8. The method of claim 7 wherein a player'"'"'s allocation is an Integrated Proportional Control (IPC) value for an allocation game v and a control game w that is determined by:

9. The method of claim 1 wherein the control game is a statistical cooperative game and an explanatory power is used as a performance measure in the statistical cooperative game.

10. The method of claim 1 wherein the allocation game is a statistical cooperative game and total effects is used as a performance measure in the statistical cooperative game.

11. The method of claim 1 wherein a manager performance attribution problem is used to generate both the allocation and control games.

12. The method of claim 1 further comprising:

approximating values in large cooperative controlled allocation games wherein coalitions selected in a control game and in an allocation game are identified by a collection of player orderings for players in the control game and a collection of player orderings for the allocation game.

13. A method for approximating a value function for players in a cooperative game based on a large number of players representing an allocation problem, comprising:

selecting a measure of precision;

determining a desired precision for approximated player values;

selecting a collection of orderings from a set of possible permutations of player orderings;

computing at least one intermediate value function based on coalitional worths generated for each selected ordering;

computing, periodically, a precision of approximations of values for players to determine if more player orderings should be generated to obtain a more precise estimate of values for players; and

computing a final value approximation for determining allocations to players when a desired degree of precision is reached or a selected computational limit is exceeded.

14. The method of claim 13 further comprising a computer readable medium having storing therein instructions for causing a processor to execute the steps of the method.

15. The method of claim 13 wherein:

a collection of player orderings is randomly generated with all players having equal probability of appearing at any position in an ordering;

an intermediate value function generated based on each player ordering is the marginal contribution of each player;

or

at least one player'"'"'s average marginal contribution is determined, or

allocations to players are based on average marginal contributions.

16. The method of claim 13 wherein:

an approximation of a weighted value is computed;

a collection of player orderings is randomly generated with a probability of a player appearing at a particular point in an ordering proportional to a ratio of its weight to a sum of weights of players not already ordered;

an intermediate value function generated based on each player ordering is a marginal contribution of each player;

at least one player'"'"'s average marginal contribution is determined;

or

at least one player'"'"'s value is based on its average marginal contribution.

17. The method of Claim 13 wherein:

an approximation of a powerpoint is computed.

18. The method of claim 13 wherein:

a standard error is a selected measure of precision; and

squared values of intermediate value functions used to compute estimated values are also computed and the estimated values squared or their sums are saved, or

sums of squared intermediate value functions and sums of intermediate value functions are used to compute sample standard errors of estimated values for players.

19. A method for approximating the proportional value for players in a cooperative game based on a large number of players comprising:

selecting a collection of orderings from a set of possible permutations of player orderings;

computing weighted marginal contributions for at least one ordering of players and one player in that ordering; and

determining allocations to players in the cooperative game.

20. The method of claim 19 further comprising a computer readable medium having stored therein instructions for causing a processor to execute the steps of the method.

21. The method of claim 19 wherein the computing step includes computing weighted marginal contributions WM_i^r(v) for each player i and ordering r in the cooperative game v with:

22. The method of claim 19 wherein the determining step includes:

determining a summation of weighted marginal contributions SWM_i(v) for a player i in a cooperative game v with;

23. The method of claim 19 wherein the determining step includes:

determining an estimated proportional value EstPV_i of a player i in a cooperative game v including N players with;

24. The method of claim 23 wherein the control game and allocation game have a same set of players and one collection of player orderings is used for both the control game and allocation game.

25. A method for decomposing joint effects in a statistical model comprising:

constructing a statistical cooperative game to represent joint effects; and

applying a cooperative solution function to determine a suitable decomposition.

26. The method of claim 25 further comprising a computer readable medium having storing therein instructions for causing a processor to execute the steps of the method.

27. A method for combining a variance decomposition of the returns of financial instrument with a factor or style analysis model of a financial instrument based on a set of benchmarks comprising computing a variance ratio for a benchmark i according to the formula:

28. The method of claim 27 further comprising a computer readable medium having storing therein instructions for causing a processor to execute the steps of the method.

29. A cooperative game resolution system, comprising in combination:

a plurality of software modules stored as data bits in memory on one or more computers with one or more processors, the plurality of software modules including;

one or more software modules for constructing a controlled allocation game, wherein allocation patterns in a control game are used to determine value allocations in an allocation game;

or

one or more software modules for approximating value functions for large cooperative games, wherein value functions are approximated to a desired level or precision using a collection of orderings from a set of possible permutations of player orderings in a the large cooperative game.