First Report on a Project Studying the Analysis of

Cooperation in Games Through Modeling in Terms of Formally

Non-Cooperative Action in a Repeated Game Context





A few years ago I gave a talk on the topic of the use of the "Prisoner's 

Dilemma" game, in a context of repetition and evolution of strategies, 

by theoretical biologists who were interested in studying the natural 

evolution of cooperative adaptations. And after giving the talk I thought 

more about the concept of studying a game by studying it as a

repeated game and through this viewpoint I got an idea of how to eliminate

all of the "verbal" complications that could become involved in the

consideration of coalitions and coalition formation.



In principle, coalitions, and specifically coalitions as considered by

Von Neumann and Morgenstern in "Theory of Games and Economic 

Behavior",  are things that could be implemented by contracts, like 

contracts in roman law. But of course a contract is quite intrinsically a 

"verbal" thing because indeed it could (or should!) be written down in words.



On the other hand, if in Nature a form of cooperation has evolved, like

with a species of insects providing fertilization for a flowering species

of plants, then the cooperation exists and is maintained, not by the

enforcement of a verbal contract but presumably by the action of "natural

selection" affecting the genetics of both species as time passes.



My idea was that in a repeated game context that the players could be

given the right to vote for "agencies" or "agents" among themselves. Thus

at a first step a player, say player A, would have the option to accept

player B as his agent. And the effect of this would be that the coalition

(A,B) would be formed (like a committee with B as chairman) without any

verbal processes occurring between A and B. Furthermore, this process

adapts to successive steps of coalescence since if another step of

elections is held then B, as the "agent" representing the coalition (A,B),

can vote to accept the agency of player C and then C will thus become the

agent representing the coalition (A,B,C).



And in this manner, with generalized "agencies" being electable, the

level of a "grand coalition" can always be reached (for a game of finitely

many players), and as a consequence of that the requisites for "Pareto

efficiency" will be available.



With regard to the actual results which have been obtained and which we

can report on now, they have been calculations of equilibria for models

and specific games where there were two or three players. The models

relate to the representation of the coalescence (or coalition formation)

possibilities in terms of the concept of agencies and the models also

incorporate some form of "reactivity" by which players can react

affirmatively when well treated by others (acting as agents) and

negatively when badly treated by other players. (And of course there 

is no bound, a priori, to the complexity of reaction patterns). But it is

somewhat plausible that with a "reasonable" level of refinement in the

modeling of reactive behavior that we can find results, in terms of

calculated equilibria of behavior, that will be plausible in relation to

appraising the bargaining/negotiation "values" for the game and also

compare calculated value vectors for it with other calculable value

vectors such as the Shapley value or the nucleolus. And the interest in

these comparisons is a good reason for studying games given as "CF games"

(which we will discuss further below) so that the Shapley value, the

classic nucleolus and the "Harsanyi nucleolus" can directly be calculated

for these games.



We have seen that there is much to do further, even on the level of

games of merely three players, to refine the modeling. We have worked

with parameters called "epsilons" which enable some smoothness and good

mathematical solutions for equilibria when they are finite and, as they

tend towards zero, limiting results that have Pareto efficiency that

becomes asymptotically perfect as "the epsilons" pass to zero limits.

(What is less clear at this stage of the research, for three players, is

that we can actually evaluate the "division of the profits" as the payoffs

to the players.)



In the continuing text below we will go into specific topics and also

into some details of calculations and of programming developed to achieve

the calculation results.





Agencies



As was mentioned above in the first paragraph, this research was

inspired by some study of the work of theoretical biologists working with

models employing the "Prisoner's Dilemma" game. When I arrived at the

idea of use of "agencies" as the means for the reduction of the potentially

quite "verbal" general concept of a coalition to a concept well suited

for studies based on analysis of the players' motivations from the

non-cooperative viewpoint of separate and independent utility measures 

I realized also that there was a connection with the studies on the topic

of "the evolution of cooperation" (Axelrod et al).



Instead of there being unlimited means by which coalitions might

actually be formed or form, dissolve, and reform, we have an election

procedure through which any player may elect to accept any other player as

his agent. And in the context of studying a repeated game we can afford to

prescribe that this election process is such that the agent is entirely

uncommitted and his election is irrevocable for each specific playing of

the game. (Of course election choices are expected to vary as the game is

repeated.)



A set of rules can be devised so that there are election stages in

each of which all of the players remaining independent (not represented by

another player as agent) have, each of them, the option of electing another

player as an accepted agent. It is natural for these rules to require

convergence so that no more than (n-1) stages of election will ever be

needed for a game of n players.



Election rules need to do something to resolve the impasse of a

situation where A votes to accept B as his agent but B simultaneously

also votes similarly for A. It is not exactly clear which rule version

handles these situations in the most preferable fashion, we have worked

with more than one variant.



When we more recently found, in the course of the use of specific model

games for calculations, that it seemed to be desirable to allow elections

to be repeated when an election had failed to result the election of any

agency power, this finding had the effect of suggesting that election

rules which had the effect that at most one agency could be elected at

any stage of the election process would be most convenient.



Concerning the general concept of transforming a cooperative game into

a form where all cooperation must be realized by means of the election of

agencies, we can remark that this is analogous to thinking of committees as

being such that all effective actions of a committee must take the form of

an action by the "chairperson" of the committee.



If one begins with a quite general CF game and then if one introduces

a game formally requiring that all coalition benefits must be attained

through the means of the action of agents who have the authority to

represent all the members of the coalition then the "agencies game"

resulting from this still has the same validly derivable characteristic

function as the original game. In essence the coalitions simply have

the same potentialities as before, but in a formal sense, for these

potentialities to be exploited, the members of a coalition would need

TO CONSPIRE on a practical procedure for electing agents successively and

finally taking the effective action of the coalition as an action of the

agent finally elected to represent all of the members of that coalition.



Remark: The agencies concept inspired the origins of this research

project on which we are reporting. And earlier we considered various

alternative specific schemes of rules for the election of agents and

agencies. Some earlier texts which describe some of these alternative

ideas are available on the author's "web page". In particular the text of

the file "agentt7c.c" gives a good picture of these ideas with alternative

varieties of election rules being considered.





The Currently Applied Election Rules for Agencies



It was initially used just for the purpose of simplifying the

format of model games and simplifying the calculations needed to find the

corresponding equilibria (for the game of the realization of "coalescence"

(and coalition benefits) through the election of agencies). But in the

process we found that a simplifying format of election rules seems to be

adequate and convenient. If in a stage of elections more than one of the

players has voted to give his acceptance to another player to become his

agent (and as if with "power of attorney") then a random process, or

"chance move", determines only one of those votes as effective, and one

of the players has elected one of the other players to be agent for him.



In our latest modeling for 3-person games we introduced the concept of

repeating an election if no player had voted to give acceptance to anyone

as his agent, but this repeated election opportunity was to be given only

with probability of (1-e4) (one minus epsilonsub4). Then we found that we

got good results mathematically by considering the limiting results as e4

(epsilonsub4) tended to the limit of zero.



In general, if there are n players of a game, with this procedure, it

would take n-1 steps of effective election to achieve the final election

of an agency for the "grand coalition" (which is the coalition including

all of the players of the game). And in general, for the realization 

of "Pareto Efficiency", it is needful that the grand coalition level 

is attained.