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The calculations that were made to find the model solutions for a range of game cases with b3 = v(1,2) varying (and v(1,3) = v(2,3) = 0) showed a problem phenomenon when b3 would increase to around 0.7 numerically. We did not understand this problem as well at first as we do now. What was beginning to happen was that the quantities a3f12 and a3f21 (or, abbreviated, af12 and af21) were vanishing and starting to become negative. (Here any negativity of these numbers, which properly describe probabilities, is only a "mathematical phenomenon", as it were.) This possibility derived from our Outline of Current Research Program

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simplification of the equations of the model from original circumstances when original variables d3f12 and d3f21 would have indicated "demands" linked with the "acceptance action" probabilities a3f12 and a3f21. For a3f12 and a3f21 to become negative, with the standard equations linking them to d3f12 and d3f21, then these later d-variables would need to become complex numbers (and thus "unreal"). We did not at first properly link these phenomena together, but these observations connect also with a set of 10 experimental games studied by a group including me and Professors Nagel, Ockenfels, and Selten of Germany and Barcelona.

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It turned out that although the experimental games could be played easily enough by "experi- mental subjects", under experimental conditions requiring that the players would necessarily have to take actions of "acceptance" in order to form coalitions and to obtain any sorts of payoffs from the playing (of the games in the experiment), it was not possible, for most the ten specified games, to actually calculate a model solution (according to the modeling that was studied in my recent paper) because the two-player coalitions did not all have comparatively small values!

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Prof. Selten had selected the 10 model games, each of which was described by three numbers thusly: b1=v(2,3), b2=v(1,3), and b3=v(1,2). If b1, b2, and b3 were all near to +1 then I would feel justified in saying "Well, this is a game that although "cooperative" in form (so that the players MAY cooperate) is really intrinsically of a competitive form in that there is potentially great competition of the alternative 2-player coalitions." Nothing is better, really, as a solution, than that one of the three alternative two-player coalitions should form and simply be accepted, by its two members, as a permanent alliance.

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We found that our mathematical model, involving solving 42 equations in 42 variables WAS effective for Game 10 of the sequence. It gave payoff results that were actually closer to the AVERAGE experimental outcome for this game than either the nucleolus or the Shapley value. (The results were actually comparatively “equalizing” in form; The payoffs to the three players were not very widely varying, comparatively).

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And we tried hard, with numerical calculat- ions, to do something about Game 9 (but we did not succeed) working with our existing model with 42 equations, etc. The striking thing was that if the three coalitions for Game 9 were weakened a little, such as by b1, b2, and b3 all being reduced by the factor of 4/5, then the 42 equations model would work for Game 9 (modified) and give numerical results.

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And what was notable in particular was that the singular phenomenon that was developing was that a3f12 and a3f21 were becoming negative (which was necessarily disruptive to the mathematics of our model). (The quantities indicated by these two numbers are: a3f12 is the probability for Player 3 to vote to ACCEPT a coalition already formed in which Player 1 has been given agency power by Player 2, and a3f21 is the dual possibility where Player 2 has agency power given by Player 1.)

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The Main Ideas For An Updated Modeling We hope to find good luck in studying games like Game 9 in the set of 10 games studied experimentally by refining our modeling (which leads to the context of the mathematical solution of the model games in the context of the special type of equilibrium that we seek to find.) (This is an “evolutionary equilib- rium” in pure strategies, where the strategies of players are described by numbers that define how they allocate utility to other players (when that becomes possible) or how they choose “demand” quantities that, in turn, control the probabilities describing their actions of acceptance.)

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(Explaining and illustrating this, a1f23 has indicated the probability of Player 1 taking the action to ACCEPT the coalition {2,3} under the circumstance of Player 2 being the elected leader of that coalition (and a1f32 would refer to a dual case where P3 is leading a binary coalition including also P2)). Here it may be well also to remark that a1f23, relating to Player 1 “accepting” the two player coalition {2,3} with P2 leading that coalition, naturally is closely linked also with u1b23r1 which indicates “the quantity of UTILITY dispensed to P1 BY the coalition {2,3} (led by P2) when that leadership has become the leadership of the grand coalition, {2,3,1} through the acceptance by P1 of the previously formed {2,3} coalition (led by P2).

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Our numerically studied model has a “demand” quantity d1f23 also associated with this possible context of the players. This was the strategy parameter through which P1 would strategically influence the probability a1f23 of his acceptance action. The mechanism had a1f23 depending both on d1f23 (chosen strateg- ically by P1) and u1b23r1 which is one of eight utility allocation numbers chosen by P2 (applying to various circumstances in which P2 becomes the final agent (or “general agent”)).

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Introducing More Demands The basic concept, inspired by Nature and the concept of evolution in Nature, was that if the three players react appropriately among themselves while playing an infinitely repeated game that these reactions may stabilize a pattern of cooperation that would otherwise not be stable. This has been achieved, to some extent, but the specific pattern of the reactions of the players (which they should display or which would best help them to naturally find suitable modes of cooperation) has not been found. Our numerically studied model had the players pro- vided with 15 “demand” parameters to control (which function to modulate the players’ negative

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or punitive reactions to behavior (on the part of other players) not well satisfying the demands. However the three players also controlled 24 parameters that specified “allocations” of utility, like, for example, the parameter u1b23r1 that was mentioned above. What seems a good idea is to (1) introduce additional demand parameters (giving to the players some alternatives relating to how to modulate their demands). And part of this would be to allow a player (or agent) to directly consider the reciprocal player (or agent) in relation to the probabilities for the basic action of acceptance (by the first party).

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It COULD BE, in conceivable elaborations of these concepts, that a player might “make demands” relating to ANY ASPECT of the behavior of any of the other players. But we have gotten the idea that it might be specifically favorable, to achieve a good model, to allow a player or agent naturally to have a demand relating to the probability of that player or agent being “accepted”, in the reciprocal direction, by whatever player or agency that the first player or agent might be confronted with.

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Illustrating the Reciprocal Demands Returning to the illustrative context described above, a1f23 and d1f23 were related parameters in our numerically studied model and Player 1 controlled the magnitude of a1f23, which describes Player 1’s probability of “accepting” the agency-led coalition {2,3} and in that modeling a1f23 was a function of d1f23 and of u1b23r1 which specifies the precise allocation of utility that Player 1 can expect whenever he “accepts” {2,3}.

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But we hope to refine the modeling by intro- ducing another sort of demand to be made by Player 1 in this context. This would be a demand that would relate, not to u1b23r1, but instead to the magnitude of a23f1, which is the DUAL acceptance probability. And thus the behavior, described by a1f23, of Player 1 (in this context) could be controlled by two “demand parameters” (acting multiplicatively). We could join to d1f23 another parameter k1f23 which would influence a1f23 to be smaller when a23f1 (the dual acceptance rate) would be observed to be smaller.

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Thus we can try a model version in which the players will have the option of supporting or encouraging a more reciprocal pattern of acceptance votes. It does seem very natural that if we are finding natural modes of cooperation for three parties which must find their cooperative equilibrium simply from being (1) able to accept agency adoptions among themselves and (2) able to react negatively to perceived (how?) “unfairness” (or inappropriateness) of actions of other players that then the found pattern of cooperative actions (the “solution”) should exhibit a substantial level of recip- rocity with regard to probabilities of accept- ance and counter-acceptance (or dual accept- ance).

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It should be remarked that this idea for a refinement of our studied model will definitely involve some challenging mathematical complications. And part of this will be that some simple functions would be replaced by implicit functions. Working With Two-Player Games This type of modeling, to study a cooperative game in terms of a model based on acceptance actions, is quite applicable to games of two players (in the form of simple bargaining games).

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One does not expect to find interesting game theory or bargaining results in relation to situations of two parties, BUT the mathem- atics of the modeling can be tested out well in this simplified context! So we probably may well test out the effect of the players being given additional demand parameters to see how that works mathematically and computationally and to see what sort of choices, at equilibrium, that they tend to make, in relation to the alternative channels for demands.

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