**
A Project Studying
Cooperation in Games
Via **

Action in a Repeated Game Context

** 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.**

** 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.
**

**can form the basis
for a game of non-cooperative form **

**to be played by the
players of the original game being translated. 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, a priori, 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 in 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. (And furthermore
the actual results of calculations seem to indicate that this convention
on elections is "asymptotically non-prejudicial" since the probability of
simultaneous votes seemed to tend towards zero while the probability of "successful"
elections tended towards +1.)**

** 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.
**

**be evolved and developed
an analogy to the mathematical methods (studies of PDE descriptions for
the air flow, etc.) that are used in weather prediction. On a materialistic-commercial
level this would naturally apply to the details of the deals in big corporation
mergers (like of Pfizer and Pharmacia recently). Or also we can hope to
get more insight relating to existing value or evaluation concepts such
as, notably, the Shapley value or the nucleolus.**

** This program
of research has led, in connection with **

**the approach to coalitions
and cooperation via the concept of agencies, to the consideration of varied
models of how the players or participants can "react" **

**to favorable or unfavorable
behavior by other players in the repeated game and to the study of varied
concepts of how the players may choose "demands" that regulate their reactive
behavior. For example, in the analogous area of the study by theoretical
biologists of the possibility of the evolution of cooperation in repeated
games of PD type, there have been found various types of "reaction instinct"
that can work to favor cooperation. Besides
the simplest "tit for tat" instinct there can be more complex variants which
may require more memory **

** I have
been studying a more complicated model (for three players) than that previously
studied (in 2001) because an expedient used in that simpler model to reduce
the number of equations and variables to be solved for seemed to lead to
problems. The payoff results seemed to depend on the comparative smallnesses
of two different parameters, both of which were supposed to be decreased
asymptotically to zero.**

and the Current Model

of elections by means of which the agency powers are elected. In the previous model there was a simpler procedure employed after a first stage of agency elections had been effective in reducing the remaining set of active players to merely two. We used a sort

of natural bargaining scheme through which competing choices, by the remaining players or agents, of utility allocations could be rationalized. However this device of simplification did not follow straightforwardly the basic concept of the agencies as players in the same sort of game interaction as that for all the players

Both versions begin with a first stage of elections where each of the three players could vote (or "choose") any specific one of the other two players to be his

**agent. In either version
these voting or choosing actions (as repeatedly taken in the repeated game
context) are described by numbers that represent, effectively, the BEHAVIOR
involved, or the probability of the action being taken when the opportunity
is presented.**

** Thus we
have an array of six numbers (and three "implied numbers") that describe
this:**

** The other
aifj variables have parallel meanings. **

**And it is sometimes
convenient to use another type **

**of symbol like n3 =
1 - a3f1 - a3f2 which is the probability that Player 3 chooses NOT
to accept either
of Players 1 or 2 as his agent; or thus this is the probability that Player
3 votes "neither" or for himself. **

**And
since the three players make their first votes simultaneously there can
be various outcomes. We have chosen a rule to simplify the process by which
there **

**is the proper response
to the votes. If the number of acceptance votes cast by the players at
the first voting opportunity is more than one then we apply a rule that
only one of these votes, chosen at random, is to be accepted, so that the
result of the election is either (1): that one of the players has elected
some other player to be his agent or (2): that none of the players has
voted to accept any other player to be an agent representing his interest.**

** We have also introduced a
convention that in case
of a failure of the players to come together at all so
that no agency has been elected that, with a certain probability, the first
stage of elections can be repeated. This idea was also used in our prior model.**

**
Election of Agencies**

in a sense, more "orthodox" with respect **to the idea of agencies
and all possibility of general cooperation is reduced to the idea of the
final election of a "general agent". So when one player has accepted another
as his agent there then remain two freely acting players and the level of
cooperation corresponding to the "grand coalition" is not realized until
one of them has elected the other as his agent.**

** But if
this final agency election fails to be realized then we can allow the existing
agency to exploit the resources of the two player coalition formed by the
two players involved. (And in a simple case, like we consider, this can
lead simply to the use of the resources specified by the characteristic
function for the coalition of that pair of players.)**

** Similarly
with the idea for the first stage of **

**elections we allow
the second stage to be repeated, with a probability of (1-e5), if neither
side has elected **

**the other side to agency
power, and the idea is that we want to study the limiting
form of the results as e5 asymptotically goes to zero.**

** Once a
"general agent" has been elected then he/she has the privilege of being
able to allocate payoffs, from the total of the payoff utility resources
available, to all of the players, including himself. Our model has simply
a resource total available of +1 which corresponds also to the Pareto boundary
of the game.**

** For each
player there are four possible ways by which he could
have become chosen as final agent. Either of two players may have initially
elected him, and this leads to two cases, or the other two players may have
initially combined in either of two ways followed by **

his being elected by the agent of that two player combination. And then as final agent he has essentially to choose a point in 2 dimensions to determine his choice of Pareto-accessible utility allocations.

** This leads
to 8 dimensions for each player and there **

**are three players so
that this leads to 24 dimensions **

**in all for the choices
made by the players when they specify allocations of utility after being
elected **

**as "final agent". These
are 24 out of a total of 39 "strategy variables" that are regarded as subject
to **

**the individual and
individually optimizing choices **

**by the players.**

** And the
other 15 dimensions of strategic choice **

**by the players correspond
to their choice options in relation to their reactive behavior (in the
repeated game). The behavior of the players that is affected by or controlled
by their reactive strategy choices is, **

**in general, their "acceptance
behavior".
**

**
The Third Stage of the Game;**

**
the Allocation of Utility
**

** When two
stages of agency election are complete then one of the original players has
become the agent for all and he "allocates" the payoffs. These are
presumed to be Pareto-efficient and so we suppose that he/she specifies three
non-negative numbers with sum = +1. The information for this is specified
by the amounts allocated to the other players and that is two numbers. This
leads to 24 strategic choices in all, of this type, for all the players.
For example, in cases of type UjBijRk player number i is
in control and was first elected by Player j and then by
player k and player i chooses to allocate the payoff of ujbijrk to player
j (and ukbijrk to Player k, but that is another case of allocation strategy
variables).
Or for example, u1b3r21 is decided upon by Player 3, who
was elected by Player 2 after Player 2 had been
chosen by Player 1 in the first stage. This is the amount allocated to
Player 1 and u2b3r21 would be the amount allocated to Player 2 (who has a
different position in the history of the elections). Player 3 **

So there are 24 "utility allocation" variables (which correspond to strategic choices by the players) and they group into the 4 categories of UjBijRk, UkBijRk, UjBiRjk, and UkBiRjk.

**
Second Stage of the Game**

**become an agent, another
player is represented through **

**this agency, and a
third player remains solo.**

** Suppose
that Player 1 now represents 2 and that **

**3 remains solo. We
call a12, short for a12f3, the probability that 1 now chooses to vote for
3 as the **

**final agent. (This
is observable behavior on the part of Player 1 in the repeated game.) And
we call af12, short for a3f12, the complimentary probability that Player
3 will vote to accept Player 1 (who already represents Player 2) as the final
agent. These classifications lead to 12 numbers, six of each type.**

** And the
12 numbers ARE NOT "strategic" choices by **

**the players involved,
rather we arrange that they are determined by "reactive behavior" regulated
by "demands" which are the actual strategic choices for the players.**

**For example a12 (or
a12f3) is specified to be A12/(1+A12) where A12 will be a positive number.
This makes a12 a positive number less than +1. And the quantity A12 controlling
a12 is specified to be **

**A12 = Exp[ (u1b3r12
- d12)/e3 ] .**

** Here e3,
or "epsilon sub 3", is intended to be made ultimately very small, as we
study the equilibria of the model. That smallness will make A12 react sharply
as d12 and u1b3r12 vary in relation to each other. The number "d12" is the
"demand" chosen (strategically) by Player **

**1 in relation to this
situation where he can vote **

**to accept Player 3
as general (final) agent or alternatively wait and hope that Player 3 will
accept
him instead (!). And what the formula takes into consideration is simply
the prospective gain or payoff to Player 1 in the case where Player 3 becomes
general agent and previously Player 1 had been elected to represent Player
2, and this is specifically u1b3r12.**

** There
are 6 demand strategy numbers of the same type
as d12 (which controls a12). And there are also six
quite analogous strategy choices like, for example, df23
controlling af23 (or a1f23). **

** In the
second stage of agency elections an acceptance can be made either by a
single player accepting the coalition led by another player (and thus that
other player’s leadership as final general agent). Or **

** "df23"
is a choice by player 1, since it is structured to control a1f23 or the
probability of ****the acceptance by Player
1, as a solo player in stage 2 of a game, of the agency of Player 2 when
Player 2 is already representing Player 3.**

** So we
have af23 = AF23/(1+AF23) or **

**a1f23 = A1F23/(1+A1F23)
with the relation of **

**AF23 = Exp[ (u1b23r1
- df23)/e3 ] being specified **

**for the control of
the acceptance behavior af23 by **

**the (strategic) demand
choice df23. Or this could **

**be called, in longer
notation, **

**A1F23 = Exp[ (u1b23r1-d1f23)/e3
] .**

**
Behavior at Stage One**

**to "demands".
The choice made is not absolutely free from any possible arbitrariness
and something more complex might also be considered appropriate.**

** Each player,
like e.g. player 2, has the option **

**of voting either for
Player 1 (behavior of probability a2f1) or for Player 3 (a2f3) or for not
voting for either of them (behavior described by n2 = 1
- a2f1 - a2f3 ). The model, in the same manner as our previous model
(of 2001), relates these behavior-describing numbers (or probabilities)
to a single demand parameter called d2 which is all of the strategic choice
by Player 2 that relates to stage 1 **

A2fj/(1 + A2f1 + A2f3) where j is

** A2fj is
specified to be Exp[ ( q2j - d2 )/e3 ]
where q2j is specified to be the calculated expectation of payoff, to
Player 2, under the hypothesis that the **

**game has passed to
stage 2 with Player 2 having become represented by Player j as his agent.
So Player 2 chooses, strategically, the demand d2 which is interpretable
as describing what Player 2 "demands" he/she should expect in gains if either,
(q21), Player 1 becomes his representing agent or, (q23), Player 3 becomes
the agent representing Player 2 in stage 2 of
the game.**

** Then the
three strategy variables, d1, d2, and d3, control the six behavior probabilities
a1f2, a1f3, a2f1, a2f3, a3f1, and a3f2 which completely describe the actual
(observable) behavior of the players in stage 1.**

**In
all we have 39 "strategic" variables in the model, 15 "demands" and 24
choices of "utility allocations".
But we replace all the demand variables, like d23 or **

** d1, by associated
controlled behavior probabilities like a23 or a1f2 and a1f3. And then
we arrive at simpler equations with most of the appearance of exponential
functions being eliminated.**

** It is
a matter of practical considerations, how **

**to find actual numerical
solutions of the resulting equations. There was experience with this problem
from work on the previous simpler model and much of the programming earlier
developed for use with MATHEMATICA for the finding of numerical solutions
could be adapted to the new model with its larger number of variables.
With the help of an NSF project assistant (AK) the 42 equations to be associated
with the 42 unknown variables deriving from the model were derived and
cross-checked. **

** At the
present time there are some areas of example**

**games for which solutions
are known and some where no explicit solution has yet been found. For example,
for **

**a game with two symmetrically
favored players who can already gain a shared payoff of 3/4 just by themselves,
no proper solution has yet been found although illegitimate solutions with
a variable assigned a **

**game-theoretically
impermissible value have been found.**

** So it
is possible that for this example that some of the behavior-describing variables
should assume extreme values or become irrelevant. And then a reformulation
of the set of equations to be solved may properly handle this example.**

** We can
illustrate the form of numerical results found for the case
on the borderline of where the **

**seems to treat the
values of the two-person coalition as "irrelevant alternatives"
and where it depends strongly on their values. This is
the case where Players P1 and P2 are symmetrically situated and have a
coalition value of 1/3 by themselves (while the coalition
of all three players is worth 1 and no other two-person coalition has
any value). **

** We can
also illustrate, with a graphics of a sort **

**first prepared last
summer, how the Pareto efficiency **

**of the game outcome
(derived from the payoffs to the players) tends to improve as e3 tends
towards zero. The effect of decreasing e3 is that the "demands" made by
the players become effectively "more precise" and that leads to the improvement
of the bargaining efficiency, as it were. These illustrative graphics were
prepared (by AK and SL) on the basis of calculated solutions for a fully
symmetric example game. We were able to find the first solutions for fully
symmetric games more easily since the effect of that symmetry was to
reduce the number of equations and variables from 42 to merely 7. **

**
****Other
researchers have also recently gotten into studies of the process of negotiating
a mode of cooperation. The work, for example, of Armando Gomes in Philadelphia
was interesting and for three person games his model was giving either the
nucleolus or the Shapley value as the evaluation result. Of course any method
which leads similarly to an "evaluation" of games becomes a natural basis
for an "arbitration scheme", using the words of Luce and Raiffa.**

** Our modeling,
as it was also with the previous model, has three parameters that describe
the values of each of the two-player coalitions. When these are small positive
numbers, say of size less than 1/3, then the nucleolus evaluation effectively
disregards them although the Shapley value gives them a moderate weighting
in its evaluation of the game.**

** The results
of the most recent computational studies using the model (in which studies
I am being assisted by the student SL working on the NSF grant for the
research project) are giving evaluation indications that are more equitably
distributed among the three players than those of the
Shapley value. But when the two person coalitions are quite weak and the
nucleolus ignores them our model does indicate some inequality in the payoffs
to the players but less what the Shapley value would indicate.**

** If two
favored players can gain in their two-person coalition 2/3 of the total
available to the grand**

**coalition (of all three
players) then the computational results from our model indicate significantly
less payoff to the two favored players than would be indicated either by
the nucleolus or by the Shapley value.**

** It seems
plausible that some refinements in modeling**

**may allow the players
to more effectively cooperate in**

**terms of the virtues
of the "alliance possibilities" **

**involving subsets of
the total set of players. I have**

**thought of some ideas
here, of varying degrees of complexity. In principle it is quite natural,
returning**

**to the analogy with
biology and evolution, that there should be some desirable behavior patterns
that encourage cooperative behavior on the part of other interacting biological
"players". We already have "demands" in the modeling and it is possible
to design things, effectively, so that the players can "demand" certain specific
forms of cooperative behavior from other players. For example, they might
be enabled to "demand" that acceptance probabilities should be more **
**in reciprocal
relationship. Since any player is competent
to be an agent and thus a leader it might be natural
for a player to "demand" that he/she should be elected
to lead with frequency like that of another player.)**

** It is
notable in the specific model currently studied that when Player
3 seems to gain more than he/she might **

**be expected to gain,
when in a disfavored position in relation to the two-player coalition values,
that part
of HOW Player 3 seems to manage this is by having a rejecting rather
than accepting behavior in relation to any elected agency representing
a coalition of Players 1 and 2.
**

** Ultimately
I came to the conclusion that it was needed to have a MORE UNIFORM concept
of "fuzziness" ****to be applied in smoothing
out the sharpness of the effects of "demand" choices. Otherwise, and this
became evident in earlier studies of two-player models, if one player would
have "sharper" demands and the other would make "duller" demands then the
player with the sharper demands would indeed become like a "sharper" bargainer
and would tend to gain an advantage in the calculated game outcomes!
The fuzziness is needed for mathematical reasons, so that derivatives of
smooth functions can be computed. But under certain circumstances it seems
that "unbalanced" fuzziness can somehow "prejudice" the game evaluation objective.**

** And ultimately
I thought of a "neutral" source of fuzziness by means of supposing a general
uncertainty about the actual boundary of the Pareto-accessible utility outcomes. **

**in the work of deriving
and verifying the payoff functions and the equilibrium equations (for equilibria
in pure strategies with parameters describing probabilities of actions
by the players). Also AK managed to effectively set up
usable programs for calculating the nucleolus. These programs had been
developed originally by S. Klauke in Bielefeld.**

** More recently
,starting in the Fall, SL has been the**

**student assistant and
has found sets of solutions for**

**varying values of the
adjustable parameters, for example for when b3 = 2/3.
**