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

• US 6,640,204 B2
• Filed: 04/06/2001
• Issued: 10/28/2003
• Est. Priority Date: 04/06/2001
• Status: Expired due to Fees

• Current Assignee: Barry E. Feldman
• Original Assignee: Barry E. Feldman
• Inventors: Feldman, Barry E.
• Primary Examiner(s): Wachsman, Hal
• Assistant Examiner(s): Raymond, Edward
• Application Number:
  US09/827,758
• Publication Number:
  US 20020198688A1
• Time in Patent Office: 935 Days
• Field of Search: 702/189, 702/181, 702/179, 702/187, 702/199, 273/460, 273/461, 703/1, 703/2, 703/17, 703/22, 705/80, 705/1, 705/35, 705/36, 705/500, 340/323.R, 340/323.B, 434/128, 434/247, 434/248, 434/249, 434/250, 434/251, 434/252, 434/253, 434/254, 434/255, 434/256, 434/257, 473/1, 473/4
• US Class Current: 702/189
• 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 joint statistical effects. A statistical cooperative game is used to represent statistical joint effects in a multivariate statistical model. Access relationships are created between players in a cooperative game and variables in a multivariate statistical model. A worth of a coalition in a cooperative game is determined, based on a multivariate statistical model and a performance measure of the multivariate statistical model. Cooperative resolution methods are applied to particular analytical procedures. Thus, the present invention 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 constructing a statistical cooperative game, comprising:

identifying a set of players for a statistical cooperative game;

identifying an access relationship between coalitions of the statistical cooperative game and elements of a multivariate statistical model, wherein a selected subset of the identified players is a coalition; and

determining a worth for selected coalitions in the statistical cooperative game based on elements of the multivariate statistical model accessible by a coalition.

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

3. The method of claim 1 wherein the step of identifying a set of players further comprises identifying a set of all available players, or identifying a set of less than all available players for the statistical cooperative game.

4. The method of claim 1 wherein the selected subset of the identified players is all of the identified players in a coalition or less than all of the identified players in a coalition.

5. The method of claim 1 wherein the step of identifying an access relationship further comprises:

identifying a plurality of elements of the multivariate statistical model;

identifying a set of coalitions in the statistical cooperative game; and

specifying an access relationship comprising a set of rules, wherein the set of rules determine for selected coalitions in the identified set of coalitions, elements that are accessible by the coalition and how the accessible elements may be used by the coalition.

6. The method of claim 1 wherein the step of determining a worth further comprises:

selecting a performance measure of the multivariate statistical model;

computing a performance measure for selected collations in the statistical cooperative game based on the elements of the multivariate statistical model accessible by a selected coalition; and

determining, for selected coalitions, a worth of the coalition in the statistical cooperative game based on the computed performance measure for that coalition.

7. The method of claim 1 further comprising determining allocations to players of a statistical cooperative game using a cooperative allocation procedure.

8. The method of claim 7 wherein determined allocations to the players of the statistical cooperative game are used to resolve statistical joint effects in a multivariate statistical model.

9. The method of claim 7 wherein the step of determining allocations to the players of the statistical cooperative game includes identifying a single such allocation for a single player.

10. The method of claim 7 wherein the cooperative allocation procedure includes a Shapley value, a proportional value, a powerpoint, a weighted Shapley value, or a log-linear value.

11. The method of claim 1 further comprising:

applying the steps of claim 1 in a recursive manner to allocate a value allocated to a player accessing a plurality of variables in a first statistical cooperative game on the basis of a second cooperative game embodying a second set of players.

12. The method of claim 1 wherein the multivariate statistical model includes continuous independent variables.

13. The method of claim 1 wherein the multivariate statistical model includes categorical independent variables.

14. The method of claim 1 wherein the multivariate statistical model includes frequency data to compute marginal frequencies.

15. The method of claim 1 wherein the multivariate statistical model includes an ordinary least squares model, a time series model, an analysis of categorical effects model, an analysis of changes in proportions model, a covariance matrix, a capital asset pricing model, an arbitrage pricing theory model, an options pricing model, a derivatives pricing model, a Sharpe style analysis model, a macroeconomic model, a price forecasting model, a sales forecasting model, or a basic or generalized Brinson and Falcher manager attribution model.

16. The method of claim 1, further comprising:

including as independent variables timing or selection factors of performance attribution dimensions; and

defining the worth of a coalition to include an incremental performance resulting from inclusion of those timing or selection factors of performance attribution dimensions accessible in a coalition.

17. A method for constructing an access relationship between coalitions in a statistical cooperative game and a multivariate statistical model, comprising:

identifying a plurality of elements of the multivariate statistical model;

identifying a set of coalitions in the statistical cooperative game; and

specifying an access relationship comprising a set of rules, wherein the set of rules determine for selected coalitions in the identified set of coalitions, elements that are accessible by the coalition and how the accessible elements may be used by the coalition.

18. The method of claim 17 further comprising a computer readable medium having stored therein instructions for causing a processor to execute the steps of the method.

19. The method of claim 17 wherein the plurality of elements of the multivariate statistical model include independent variables.

20. The method of claim 19 wherein a specified access relationship between coalitions and independent variables is one-to-one transferable and every player in the statistical cooperative game has primary access to an independent variable, each independent variable has a primary access relationship, and the variables accessible by any coalition are exactly those accessible by its member players.

21. The method of claim 19, wherein a specified access relationship between coalitions and independent variables includes a first set of basic independent variables that have primary access relationships with individual players and a second set of interaction independent variables, each of which is accessible by any coalition whose members include all the players that have primary access to the basic independent variables used to construct the interaction independent variable.

22. The method of claim 19 wherein independent variables accessible by a coalition may be used to construct submodels of the multivariate statistical model that are based only on accessible independent variables.

23. The method of claim 19 wherein access to an independent variable allows a coalition to use the estimated coefficient for the independent variable based on the complete multivariate statistical model including all independent variables, in a computation of a performance measure.

24. The method of claim 17 wherein a specified access relationship between coalitions in the statistical cooperative game and elements of the multivariate statistical model includes a property that all variables accessible by any first coalition of players are also accessible by any second coalition including all the players of the first coalition.

25. The method of claim 19, further comprising applying a procedure of returns-based style analysis, wherein:

a dependent variable is a returns time series for a financial security;

independent variables include return time series for a set of asset class benchmarks;

an asset class benchmark has a primary access relationship with a single player;

a measure of explanatory power is a R² coefficient;

submodels are constructed for sets of independent variables corresponding to all coalitions in the game;

or

a proportional value of the dual game is used determine allocations to players.

26. A method for determining a worth for selected coalitions in a statistical cooperative game, comprising:

selecting a performance measure of a multivariate statistical model;

computing a performance measure for selected collations in the statistical cooperative game based on elements of the multivariate statistical model accessible by a selected coalition; and

determining, for each coalition from the selected set of coalitions, a worth of a coalition in the statistical cooperative game based on the computed performance measure for that coalition.

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

28. The method of claim 26 wherein the performance measure is an unadjusted R² statistic, an R^2* statistic, a predicted value of a dependent variable, a value of a log likelihood function, a variance of a forecasted observation, or an out of sample mean square error.

29. The method of claim 26 further comprising constructing a second statistical cooperative game based on a same set of players by constructing a dual game to the statistical cooperative game.

30. The method of claim 26 wherein a worth is determined for every coalition in the statistical cooperative game.

31. The method of claim 26 wherein a worth is determined for less than every coalition in the statistical cooperative game.

32. The method of claim 26 wherein the steps of selecting a performance measure and computing the performance measure for a coalition do not utilize elements not accessible by a coalition.

33. The method of claim 26 wherein the step of constructing the statistical model includes constructing the statistical model for a complete set of all elements.

34. The method of claim 26 wherein the step of constructing the statistical model includes constructing the statistical model for less than a complete set of all elements.

35. The method of claim 26 wherein the step of computing of a performance measure includes using coefficients estimated for independent variables accessible to the selected coalitions.

36. A method for allocating a worth of a coalition in a cooperative game on a multiplicative basis, comprising:

generating a second cooperative game from a first cooperative game by setting a worth of a plurality of coalitions in the second cooperative game to a logarithm of the worth of a same coalition plus a constant;

applying a cooperative allocation procedure to the second cooperative game; and

creating an allocation for a player in the first cooperative game from an allocation in the second cooperative game by applying an antilog to a value allocated to a player in the statistical cooperative game and subtracting a constant.

37. The method of claim 36 further comprising a computer readable medium having stored therein instructions for causing a processor to execute the steps of the method.

38. The method of claim 36 wherein the value allocated to a player i is determined by a log-linear value:

${\mathrm{LL}}_i<br><br>\left(v\right)=\exp <br><br>\left(\underset{S\ni <br><br>i}{\sum <br><br>}<br><br>\frac{\left(n-s\right)!<br><br>\left(s-1\right)!}{n!}<br><br>\ln <br><br>\left(\frac{1+v<br><br>\left(S\right)}{1+v<br><br>\left(S<br><br>\backslash <br><br>\stackrel{\_}{i}\right)}\right)\right)-1,$

wherein the summation is over all coalitions S that contain player i, n is a number of players in the cooperative game, and s i s a number of players in a coalition S.

39. A method to estimate a total effect of a basic variable on an individual observation, set of observations, or forecast value in a multivariate statistical model with interaction variables, comprising:

constructing a multivariate statistical model with basic variables and a plurality of interaction variables derived from the basic variables;

identifying an access relationship between coalitions in a statistical cooperative game and elements of the multivariate statistical model, including the basic and interaction independent variables;

constructing a cooperative game based on the access relationship; and

applying a value function or other allocation rule to the cooperative game to attribute all effects among players in the game.

40. The method of claim 39 further comprising a computer readable medium having stored therein instructions for causing a processor to execute the steps of the method.

41. 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;

a player module for identifying a set of players for a statistical cooperative game,

an access module for identifying an access relationship between coalitions of the statistical cooperative game and elements of a multivariate statistical, wherein a selected subset of the identified players is a coalition, and

a worth module for determining a worth for selected coalitions in the statistical cooperative game based on elements of the multivariate statistical model accessible by a coalition or for determining a worth of a coalition in a cooperative game on a multiplicative basis; and

one or more databases for storing cooperative game data.