Simultaneous Resource Accumulation and Payoff Allocation : A Cooperative Fuzzy Game Approach

We develop a simultaneous resource accumulation and payoff allocation algorithm under the framework of a cooperative fuzzy game that builds on our earlier work on the role of satisfaction in resource accumulation and payoff allocation. The difference between the twomodels lies in the fact that while focus wasmore on getting an exact solution in our previousmodel, the negotiation process in the currentmodel accountsmore for the role of the intermediate stages.Moreover we characterize our solution using two properties: asymptotic fairness and efficiency. Our model includes a suitable penalty function to refrain players from unreasonable demands. We focus on real life situations where possibly one or more players compromise on their shares to ensure a binding agreement with the others.


Introduction
In the literature, cooperation among self-interested players under binding agreements has been well explained by the theory of crisp cooperative games.In [1][2][3] it is shown that players' satisfaction has the ability to set up a solution concept in a cooperative game which is easily computable through an algorithm.It is therefore important to take account of how coalitions are formed vis-à-vis and how the worth is allocated among participating players so that they are individually satisfied up to a desired level.
In this paper, we propose a dynamic process of accumulating resources from players and of finding a payoff allocation simultaneously under a cooperative fuzzy game theoretic environment.The preliminary works are done in [3].Here we enhance the same model incorporating (a) the role of players in the intermediate steps of the negotiation process and (b) the notion of a penalty function that restrains the players from making unreasonable demands.
Consider, for example, the role of Uber and Ola cab services in different Indian cities.Both these companies acquire vehicles from the local vehicle providers of the cities on lease and offer them a portion of the profits accrued from the customers on per day basis.It follows that the cab services accumulate resource (in terms of number of vehicles from different service providers), generate some profit, and finally allocate this profit among the stack holders.It may be the case that the local vehicle providers have their shares in both the companies that results in different levels of satisfactions in terms of their trade relations with these companies.Retention becomes a big concern for both the companies.Therefore they need to satisfy the vehicle providers by providing sufficient opportunity to both generate and share the profit.More precisely let us consider a hypothetical situation where three agents  1 ,  2 , and  3 in their respective territorial areas collect monetary resources from the customers and provide this accumulated resource for an investment firm.The firm in turn invests the whole amount in shares of various companies (finitely many) in the market and rewards the agents according to their performances.The agents take notes on how their resources are being invested in different companies.This would determine how much payoff they would get after the firm profits from such investments.
Here the manager of the firm acts as the mediator who Advances in Fuzzy Systems takes care of all the resources of the agents.The payoffs to the agents are usually measured through their resource collection capacities, whereas, in addition to weighing on such straight performances, many organizations provide extra incentives adopting background corrections.This may include the geographical disadvantages the agents have to comply with in their territorial areas, their prospects towards future expansion of business, team working capabilities, and so on.Such incentives essentially gear up the satisfaction levels of the agents.However it is not found in the literature whether any standard procedure is followed in provision of such incentives.Moreover most of the business organizations follow the principle "if you perform well you get betteroff."Note that performance and satisfactions can never be universally standard.This motivates us to pursue the present study.Similar works can be traced back to [4][5][6] and so on.
In [7] a dynamic payoff allocation method that converges to a specific allocation under a given set of external restrictions is designed.Similar models of frequency allocations in wireless networks through a game in satisfaction form are found in [8] where payoff to a player accounts only for her satisfactions.However none of these studies uses the notion of fuzzy cooperative games as a tool to address their problems.For similar studies we refer to [4,9,10] and so on.
In our previous model [3], the problem of simultaneous resource and payoff allocation among participating players by a mediator is discussed under a dynamic setup.It shows that an exact resource versus payoff allocation matrix evolves as a solution to a minimization problem at each stage of the negotiation process.However we observe that here the emphasis is put on how the exact solution (as a part of the limiting process) can be obtained without accounting for the intermediate solutions.In real life situations one needs to consider time and monetary constraints that do not allow the negotiation process to last longer.Therefore, in our present model we focus on the fact that the players get an optimal solution with the least possible time and money.We provide an axiomatic characterization of the exact solution which is based on two very natural axioms: efficiency and asymptotic fairness.We provide an example at the end of the paper to highlight this issue.
Let  = {1, 2, . . ., } be the set of players (agents) which is known as the grand coalition.Any subset of  is called a crisp coalition.In a crisp coalition, a player can give either full participation (investment in our case) in the coalition with her complete resource or no participation (no resource).The collection of all crisp coalitions of  is denoted by 2  .A crisp cooperative game is a pair (, V) where V : 2  → R + ∪ {0} is a real valued function known as the characteristic function such that V(0) = 0.If the players' set  is fixed, the cooperative game is denoted by the function V.For each crisp coalition  ∈ 2  , the real number V() is known as the worth of the coalition  (or profit incurred from ).On the other hand, if a player needs to participate in more than one coalition simultaneously, with her resources (or power) in hand, it is possible to provide only fractions of her full resource (power) for those coalitions.Such coalitions are called fuzzy coalitions.A fuzzy coalition is nothing but a fuzzy set of , represented by an -tuple, where its th component is the membership degree (or fraction of the power) of player  ranging between 0 and 1.A crisp coalition can be realized as a special type of fuzzy coalition where the degree of participation of any player in it is either 1 or 0. In the literature Aubin [11], Butnariu [12], and Branzei et al. [13] have well developed the theory of cooperative fuzzy games and justified the fuzzification in terms of the players' membership degrees in a coalition.In both crisp and fuzzy environment, it is quite necessary to determine how a fuzzy coalition structure is formed as well as how a suitable payoff distribution is proposed to the players accordingly.Solution concepts for cooperative games are found in [11][12][13][14][15][16][17].In [1], a new solution concept that evolves as a result of a dynamic negotiation process is defined.It is termed as the payoff allocation.In [2], a process of resource allocation of players is obtained in different cooperative actions with the formation of a fuzzy coalition structure.The key objective in [1][2][3] is to investigate the influence of individual satisfactions upon a payoff and resource allocations (i.e., how it can be used to arrive at a suitable payoff and resource distribution over time).The process of resource allocation (equivalently resource accumulation) is made here synonymous with the formation of a fuzzy coalition.It follows that their model is equivalent to solving an -person cooperative game with  distinct fuzzy coalitions.
The goal of our present study is to provide a more developed and systematic treatment of satisfaction levels as a basis for negotiation among rational agents, who are capable of participating in different fuzzy coalitions with possibly varied rate of memberships simultaneously.We introduce the notion of a penalty to restrict the irrational demands of the players.Our present model is seen to be efficient than the one discussed in [3] in situations where the allocation process does not continue for long rather than stopping at some intermediate stage as part of a trade-off.We modify the variance function of [3] and show that at an intermediate stage our solution is more efficient than that of the earlier model.
The allocation process (Figure 1) goes exactly in the same way as that in [3].Initially, the mediator would accumulate budgets/resources from the rational players.The mediator and the players would jointly determine the number of possible coalitions.Then the players would announce their expected total budgets/resources for each coalition.The mediator finds the optimal coalition structure and the optimal total budget allocation in such a way that the sum of all budgets allocated for all coalitions of the optimal coalition structure is equal to the total budget in her hand.
In the next stage of the allocation process, the mediator proposes the resource and payoff allocations simultaneously for each coalition of the optimal coalition structure which we call a solution here.The rational players would provide their satisfaction degrees in each coalition according to their own investment and payoff satisfaction functions.Based on this information, the mediator would update her belief and propose the next solution.The process continues until a stopping condition is reached.Thus the mediator would propose successive offers (solutions) to the players judging on  their reactions to the previous proposal.Similar to what was proposed in [1][2][3] we construct a stopping rule and propose the process of updating the belief of the mediator by use of a pair of suitable functions towards the possible reactions of the players upon different offers of resource allocations and payoff allocations.We call them, respectively, the approximate investment satisfaction and approximate payoff satisfaction functions.A variance function is constructed to evaluate the closeness among the degrees of investment satisfaction and payoff satisfaction of the individual players in a coalition over a solution.If the variance of a solution is below a certain threshold to be determined by all the players collectively, then it would be considered as a possible trade-off solution to the problem.Situation may arise where the variance does not converge further after a particular threshold which is still an open problem and we keep it for our future works.
An exact solution is a solution whose variance is zero, see [3].Therefore for an exact solution all the investment and payoff satisfactions are equal in each coalition.The process of proposing new allocations at successive stages leads to an exact solution.We assume that every player keeps her satisfaction functions unknown to the mediator and other players.The negotiation strategy is designed so that the mediator would propose only offers (possible solutions) for Advances in Fuzzy Systems which the variance would be minimum at each stage of the negotiation process.In the negotiation process, each rational player has a single motive: to maximize her individual payoff, which is well represented by some monotonic increasing functions characterizing the fuzzy sets of their satisfactions.Since, negotiation requires a player to be considerate about the desires and views of all the other players.An appropriate negotiation process can restrain the players from making irrational demands while rewarding ones who are willing to work together by forming coalitions.
The remaining part of the paper is designed as follows.In Section 2, we build the theoretical framework of our proposed model and prove the existence of a better offer as a solution to our model.Section 3 deals with the notion of a penalty function to protect the negotiation processes from irrational demands.Some hypothetical examples are presented to discuss our model in Section 4. In Section 5, we present the concluding remarks.This section builds mainly on the notions and results discussed in [3].We assume that for every  ( = 1, 2, 3, . . ., ), the amount of resources available for agent  is   ≥ 0 (this can be time, money, etc.).In order to make the model simple we take the resource inputs and the payoff outputs of each player as real numbers.Each player  can choose to invest in a joint project any portion   ≤   ∈ R of her total resource.We call this joint project a fuzzy coalition.The total resource allocation is represented by a nonnegative vector R = ( 1 , . . .,   ) ∈ R  , and possible coalitions are identified with the vectors that are (coordinatewise) smaller than R. By what is called an abuse of notation we shall represent the sum ∑  =1   by  here.
Thus formalizing the notion, we have the following: for every nonnegative vector R ∈ R  , let (R) be the box given by The point R is interpreted as the "grand coalition" in fuzzy sense, and every r ∈ (R) is a possible fuzzy coalition while 0 ∈ (R) is the zero vector where all the players put zero resource.For every R ≥ 0 : R ∈ R  , a cooperative fuzzy game is a pair (R, V) such that Since infinitely many fuzzy coalitions with limited resources of the players are not practically useful, let there be only  possible fuzzy coalitions ( < ∞).A fuzzy coalition structure S  (, ) = (S  1 (), S  2 (), . . ., S   ()) with respect to the grand coalition R (or equivalently ) is an -vector whose components S   (), 1 ≤  ≤ , with  players are called fuzzy coalition variables.Thus a vector r ∈ (R) is a fuzzy coalition in the fuzzy coalition structure S  (, ) if S   () = r for some  : 1 ≤  ≤ .
Note that if V : (R) → R + ∪ {0} is continuous and all the resources are of same kind, then V depends only on the total amount of resources  of a coalition r ∈ (R) instead of different distributions: that is, V is constant on every set {( 1 ,  2 , . . .,   ) | ∑  =1   = } for each  ∈ R. For example, if resources are considered in monetary units, then V being symmetric in all variables generates a unique function  : R → R + ∪{0} such that V =  where  : (R) → R is defined for every r ∈ (R) by (r) = ∑  =1   .Thus with an abuse of notations we can use V and  alternatively and hence represent V(r) by V(∑     ).For every  ∈ {1, 2, 3, . . ., }, let the maximum resource that can be accumulated in the th fuzzy coalition be   .In view of the above discussion a fuzzy coalition structure S  (, ) can therefore be denoted by an -tuple of real numbers ( It follows that a resource investment matrix is a fuzzy coalition structure. Definition 2 (see [3]).A solution (x, y) of a cooperative game V with  players and  fuzzy coalitions is an  ×  bimatrix, where the coordinates of the (, )th entry (  ,   ) represent the respective resource investments and payoff allocations to player  in the th coalition, satisfying the following four conditions: (i) 0 ≤   ≤   , 0 ≤   ≤ V(  ),  = 1, 2, . . ., ;  = 1, 2, . . ., .
Let S , (V) denote the set of all solutions (x, y) with respect to a cooperative game V with  players and  fuzzy coalitions.The column vectors x  and y  ( = 1, 2, . . ., ) corresponding to a solution (x, y) represent, respectively, the resource investment and payoff allocation vectors for the th coalition.
1.1.Formation of the Cooperative Fuzzy Game.Initially, player  ∈  = {1, 2, . . ., } would submit her budget   to the mediator.The mediator and the players jointly determine an optimal coalition structure; see [2]. 1  The mediator offers fractions of resources among the players and offers a possible investment vector to invest in a joint project (that is to form a fuzzy coalition).The players would react by announcing their degrees of satisfactions.Thus, we associate a satisfaction function for each player for such resource investment and call it the investment satisfaction.
Thus the investment satisfaction function of player  in the cooperative fuzzy game V, denoted by   V , is defined as follows (see [3]).Definition 3. Let   be the total resource of the player  ∈  in a cooperative game with fuzzy coalitions V. Then the function   V : R → [0, 1] is said to be an investment satisfaction function of player  if the following hold: V is continuously differentiable and strictly monotonic increasing in [0,   ].
Probable physical significance of the above assumptions as given in [1] is that each player  is keen to invest her whole resource   .Her degree of satisfaction is therefore zero if she has no investment at all and one if she has full investment there.Moreover, it is natural to expect that satisfaction of any player increases continuously with respect to her investment.Furthermore, every player tries to increase her resource investment in a coalition, so the investment satisfaction functions are convex.
The payoff satisfaction function for player  over her payoff allocation in the cooperative fuzzy game V, denoted by   V , is defined as follows.
where  − is the total resource in some coalition where player 's investment is zero and each player , such that  ̸ = , invests   in it.One can interpret   as the fuzzy version of the marginal contribution of player .Then the function   V : R → [0, 1] is said to be a payoff satisfaction function of the player  if the following hold: V is continuously differentiable and strictly monotonic increasing in [0,   ].

(c) 𝑃 𝑖
V is convex or equivalently the derivative of Probable physical significance of the above assumptions as given in [1] is that each player  has a degree of satisfaction for any reward ranging from zero to   , where zero is the least reward player  can assume and   the maximum.Moreover, degree of satisfaction increases if the reward increases from 0 to   .So the function   V is considered as continuously differentiable and also strictly monotonic increasing in [0,   ].Again in any coalition the players mostly try to increase their payoffs by decreasing their satisfaction degrees upon an offer.Thus Assumption 3 reflects the choice of a particular satisfaction function that tries to address the individual preferences of each player.
Remark 5. Since the range of investment satisfaction and payoff satisfaction functions in Definitions 3 and 4 is normalized to [0, 1] the implicit assumption we make here is that the satisfaction of each player is a cardinal utility that admits an interpersonal comparison.
Definition 6 (see [3]).A solution (x, y) of a cooperative game V is said to be an exact solution if all the players in every fuzzy coalition are equally satisfied with their resources and payoffs; that is, Definition 7 (see [3]).A solution (x, y) of V is said to be an approximate solution if there exist  and ( ̸ = ) such that for some  ∈ {1, 2, . . ., }.
Let H be the history over which the mediator makes proposals to the players.Given a time (stage)  ∈ H and the set S  , (V) of solutions (x  , y  ), let us define the following sets [3]: where  V (x   ) = ∑  =1   V (   )/ and  V (y   ) = ∑  =1   V (   )/.For each  ∈ H, let us call the solution (x  , y  ) ∈ S  , (V) a dynamically evolved solution (DES) at time .

Our Model
Definition 8.In order to obtain a better solution from the previous solution, we define the variance function var : S , (V) → R as follows: where  = (  V (  )  V (  )− V (x  )  V (y  )) 2 and  V (x  ) and  V (y  ) are as given by Eq. ( 2).
Note that in (3),  relates to the satisfactions of the players with their resources and payoffs at each stage.In our model in [3], effect of  was implicit and could only be realized at the last stage of the exact solution searching procedure.In the intermediate stages the investment and the payoff satisfactions were assumed to be independent of one another.

Advances in Fuzzy Systems
However in reality this is more of a strong assumption as satisfactions cannot be treated in isolation.Moreover, we have already explained that due to time and monetary constraints it is unlikely to allow the allocation process to iterate for long to reach the exact solution and rather a trade-off is sought to arrive at some optimal solution.Thus  suggests that optimal solutions at intermediate stages should also account for such interdependence between the investment and the payoff satisfaction functions.Definition 9. A solution (x, y) is said to be a better approximate solution over another solution (x  , y  ) if var(x, y) < var(x  , y  ).

Negotiated Allocation Strategies.
We assume that, in our allocation model, the negotiation is governed by the negotiation strategies adopted by the mediator.These strategies determine how the mediator identifies suitable solutions and how the players evaluate those solutions in the light of their own interests.The mediator would propose solutions at each stage/time of the negotiation process until a solution is accepted by the players with equal satisfactions within a coalition using exact solution searching strategy [3].Alternatively the mediator may suggest the players to adopt a trade-off or an optimal solution searching strategy [3] which we discuss in the following briefly.
In the allocation process the mediator does not have any initial information about the investment and payoff satisfaction functions of the players; she will only update her belief on each player at each succeeding stage using the lower and upper bounds of those satisfactions.At stage  of the allocation process, these bounds can be obtained for player  ∈  in the th coalition by defining two approximate functions (see [3] for more details); namely,    : R → R and    : R → R.These functions are simply linear approximations (from below) of the actual satisfaction functions of the players for the next offer at stage  + 1 and are obtained by joining the pairs of points [(0, 0), (   ,   V (   ))] and [(0, 0), (   ,   V (   ))], respectively.Thus we have Similarly for x   ∈   2 denote Finally for x   ∈   3 denote In a similar manner, denote for For y   ∈   2 denote Finally for y   ∈   3 denote Thus the expected proposal (x * , y * ) that benefits all the players can be defined as follows.
Definition 10.Assuming that the mediator offers a DES (x  , y  ) to the players at stage  and that all the players subsequently announce their investment and payoff satisfaction degrees, the expected better resource allocation (x * , y * ) for the proposal at stage  + 1 is defined as where Figure 3 and the set of constraints  is given by Here  is the sum of the squared differences of different satisfaction levels of the players using approximate satisfaction functions in each fuzzy coalition.If at stage , the proposed DES (x  , y  ) is such that, for some player , any one of   1 ,   2 ,   1 , and   2 is empty, then the exact solution searching process fails.In such case, if there exists an all player-accepted trade-off threshold  such that var(x  , y  ) < , the current solution is accepted as an optimal solution.Otherwise the allocation process stops there and we say that the process ends with conflict.
Step 1.All the players will submit their resource   ,  = 1, 2, 3, . . ., , to the mediator.Also, the players will construct their investment as well as payoff satisfaction functions independently using the cooperative game V.
Step 2. The mediator along with the players will determine the optimal coalition structure { 1 ,  2 , . . .,   } where   is the total budget for the coalition .
Proof.We have to prove that there exists a better DES at stage  + 1 over the one at stage .This is achieved in two steps.In Step 1 we prove the existence of such a solution to the game, and then in Step 2, we show that this solution is better than the previous one.
Step 1 (existence of a solution).This part follows directly from Theorem 1 in [3] and so is omitted here.
Step 2 (there is a new DES better than the previous one).Here we are using the fact that ∑  =1   = ∑  =1   , along with the following: (i) for some player  and some for some player  and some Similarly, one can find a y ∈ R  × R  using the following facts: Thus we have Therefore, (x, y) is a better solution than (x  , y  ), so either we can denote (x, y) by (x +1 , y +1 ) or we can find better DES (x +1 , y +1 ) which will minimize var(x, y).This completes the proof.
Theorem 13.If at each stage , for each player , none of the sets   1 ,  2   ,  1   , and  2   is empty, then the process of obtaining a better DES converges to the exact solution.
Proof.The proof proceeds exactly in the same way as of Theorem 2 in [3] and so is omitted here.

Axiomatization of the Exact Solution
It follows from Theorem 13 above that the notion of an exact solution represents fairness and egalitarianism among players and the mediator is assumed to minimize the deviation from the fairness criterion.It is natural to ask: What should be a player's ultimate interest?Investment satisfaction or payoff satisfaction?The negotiated allocation strategies are designed so that the deviations in both investment and payoff satisfactions are minimized simultaneously.Thus a formal characterization of the solution to rationalize these strategies is equally important.In this section, we provide an axiomatic characterization of the exact solution with the axioms of efficiency and asymptotic fairness as mentioned in Section 1.In what follows, we provide the formal descriptions of these axioms.Definition 14.A solution (x * , y * ) to the cooperative fuzzy game V ∈   (R) is called asymptotically fair if there exists a sequence of DES (x  , y  ) for  ∈ H and a continuous function It follows form Definition 14 that, when a solution is asymptotically fair, some kind of equitability among the players is preserved.The continuous function  determines the type of equitable condition to be met by the solution.
Asymptotic Fairness (AF).A solution (x * , y * ) to the cooperative fuzzy game V ∈   (R) is asymptotically fair.Efficiency ().For (x, y) ∈ S , (V) of V ∈   (R) it holds that (18) Note that the efficiency axiom  conforms with our basic assumption in the current model that accounts for accumulation of resource and allocation of payoffs together.Thus we have the following theorem.Theorem 15.There exists an efficient and asymptotically fair solution to any game V ∈   (R).It is uniquely determined by the continuous function  given by (17).In particular when  ≡ var given by (3), it is the exact solution that is both efficient and asymptotically fair.
Proof.The proof follows from Lemma 11 and Theorems 12 and 13.

Penalty
Let us now introduce the notion of penalty to the negotiation process.This makes the allocation model more realistic.Usually, it happens that some players expect unreasonably high payoff by contributing merely in a coalition.This would render a never ending process of negotiations with little progress.In order to avoid such situations, the mediator may impose some penalty on the players who ask for unreasonable payoffs by showing unfair satisfaction against offers.We start with few definitions.Definition 16.An investment satisfaction function for player  ∈  is said to be normal if its curve of description is a straight line, passing through the points (0, 0) and (  , 1), with   ̸ = 0. Thus, the equation of a normal investment satisfaction function for any player  is given by If the negotiation stops here and (x  , y  ) is the required solution of the problem, using (a) to (e), we see that the penalty of player  1 is ∑ 2 =1  1 ( 1 ,  1 ) = 0 + 0 = 0 and penalty of player  2 is given by ∑ 2 =1  2 ( 2 ,  2 ) = 0.622996 + 0.622996 = 1.24599.

Conclusion
This paper illustrates a dynamic allocation process to solve the problem of resource investment and the corresponding allocation of payoffs among the players in a cooperative environment contemporaneously.Since players may be involved in different projects at a time and are able to provide only fractions of their resources, we adopt a cooperative fuzzy game theoretic approach.Convergence in the proposed model is restrictive and each player is endowed with two satisfaction functions: one for resource investment and the other for payoff allocation.Combining these functions to have a common satisfaction function and obtaining a more general formulation for convergence are the topics of our future study.

Figure 1 :
Figure 1: Flowchart of the allocation process.
will repeat Step 4 to get an exact solution with variance 0 or optimal solution with variance less than  (nonzero).Let the DES at stage  be (x  , y  ) = [(   ,    )] × such that for each player , none of Theorem 12. Let V ∈   (R).At stage , let (x  , y  ) = [(   ,    )] × be a DES which is not exact, and for each , none of the sets Definition 17.A payoff satisfaction function for player  ∈  is said to be normal if its curve is a straight line, passing through the points (0, 0) and (  , 1), with   ̸ = 0.Advances in Fuzzy SystemsLet at some stage  the proposal offered by the mediator be(x  , y  ) = ( (2, 7.27168) (2, 7.27168) (4, 10.7283) (4, 10.7283) ) .