The article examines the problem of the efficient allocation of resources in conditions of incomplete information concerning the parameters of agents' utility functions. Through business game the results are modeled and compared in conditions of incomplete information concerning the agents' utility functions.We experimentally prove the inexpediency of information distortion of the agents' effectiveness when using a nonmanipulative distribution mechanism in a multi-step game. behavior models; utility functions; nontransferable utility; fuzzy logic The problem of effective resource allocation occurs in various applied problems [4]. If the resource value is limited and the participants interests do not coincide, a conflict situation arises. Interaction of participants in this case can be considered as a game. The description of several agents interaction includes the following parameters: • the multitude of agents; • set of permissible actions; • agent preferences (he/she is assumed that each agent is interested in maximizing his profits); • awareness of agents (at the time of making decisions about the chosen action); • the order of functioning (the sequence of actions). These parameters set the game. The game purpose is to define the multitude of active agents' actions. It means finding an equilibrium situation.
Let us consider the problem of distributing the resource R between n players. R be a distributable resource; N is the number of ( ) = −
2, > 0, > 0 is the utility function of the i-th player.
Obviously, the player will get the maximum profit at the point ∗ = . show that ( 1, 2, … ) reaches a maximum at the point ( 10, 20, … 0), where
If ( 1, 2, … ) =
∑ =1 = , then it is easy to ∑ ≠ = … − + … , = ̅̅̅,̅̅, = ̅̅̅,̅̅. If ∗ ≠ then the i-th player will be interested in increasing his profit.
As a mathematical model of the described interest conflict situation, we will use the business game for resource allocation R between n players with a reverse priority mechanism. At each step of the game the participant makes an request si to the resource. The request is satisfied by the Resource Allocation Center in the volume ∑ =1
( ) =
, = ̅1̅̅,̅̅, where = ( ∗).
It should be note that the resource distribution is based on the knowledge of the values = ( ∗) of each of the participants. In a sense, Ai can be interpreted as the utility limit of the i-th player. Let us suppose that the true values of Ai are not known to the Center (the cost factor ai is known only to the player) and for the distribution of the resource the players themselves inform the Center of the value Ai. In this case, the player has the opportunity to exaggerate, downplay the limit of its usefulness or to convey its true meaning. Also, at each step, players report the value of the required resource, which is adjusted by the players in order to obtain the desired amount.
How will the distribution of the resource change in conditions of incomplete information of the Center about the usefulness of the players? Is it possible in such conditions to maximize the profit of an individual player and the total utility of the pl ayers? Is it profitable for participants to hide the true meaning of the limit of their usefulness?
information levels of the players parameters conditions. different participants' behavior models; resource needs, using different participants' behavior models; - to conduct a comparative analysis of the results.
The purpose of this study is to compare the participants profit size in a business game on the resource distribution in different - to conduct a computational experiment in incomplete information conditions about the needs of players in resources, using - to conduct a computational experiment in incomplete information conditions about the players target functions and their
The BRM [
αi is the degree of satisfaction of the request; N is the proportion of players with αi ≥ 1.
(
+1) = +1 = +1 ∑ =1 − +1 ∗
= ∗, ( − ∗) ∑ =1 − = ( ) ∗ The rules base, which gives an assessment of the attractiveness of the player's actions, consists of the possible actions: - to increase the request, - to lower the request, - not to change the request.
attractiveness is great. change the request is great. attractiveness is great. bid increase is great.
R1. If the degree of the request satisfaction αi is small and the players share N is low, then the declining of the request
R2. If the degree of the request satisfaction αi is small and the players proportion N is high, then the attractiveness not to
R3. If the degree of the requests satisfaction αi is close to 1 and the players share N is low, then the declining of the request
R4. If the degree of the requests satisfaction αi is close to 1 and the share of players N is high, then the attractiveness of the
request is great.
increase is great.
R5. If the degree of the request satisfaction αi is large and the players share N is low, then the attractiveness not to change the
R6. If the degree of the requests satisfaction αi is large and the share of players N is high, then the attractiveness of the bid
As a result of FLM, the evaluation λ[
Relative location of ∗ means that the optimal resource values in different functions have the same deviation from equal distribution. In this caseplayers have the same chance to be winner.
Behavior model means that players use special rules for their actions. 4. Results and Discussion 1 2 3 4 7 8 9
10 5
6
BRM BRM BRM BRM BRM BRM FLM FLM FLM FLM FLM FLM c d х1 х2 х3
1 2 3 4 5 6 7 8 9 10
In all cases, the deviation of the obtained resource from the optimal individual indicator is approximately the same.
Table 2 shows the relative deviations of the resource obtained by agents in cases of reliable reporting of information on effectiveness, overestimation of the first agent effectiveness, underestimation of the first agent effectiveness in the first step. a is the players provide reliable information about the maximum of their profits; b is the players overestimate the value of their maximum profit; c is the players underestimate the value of their maximum profit; d is the players distort information about the maximum of their profits. -0,07 №1 0,12 0,16 №2 0,25 0,12 0,00 №2 0,12 0,16 №3 0,43 0,00 -0,15 №3 0,12 0,16 № 1. In the first step all players provide reliable information about their own effectiveness and the amount of the required resource. In the next steps distorting the value of the resource request is distorted in accordance with the chosen behavior model.
№ 2. In the first step the player 1 overstates the information on its own efficiency by 20%, other players provide reliable information about their own effectiveness and all players report reliable information about the amount of the required resource. In the next steps distorting the value of the resource request in accordance with the chosen behavior model.
№ 3. In the first step the player 1 understates information about its own efficiency by 20%other players provide reliable information about their own effectiveness and all players report reliable information about the amount of the required resource. In the next steps distorting the value of the resource request in accordance with the chosen behavior model.
Table 3 shows the resources relative deviations obtained by agents in cases of reliable reporting of information on effectiveness, overestimation of the first agent effectiveness, underestimation of the first agent effectiveness in the tenth step.
х3 0,17 0,17 0,16
The results of the calculations presented in the tables 2, 3. The information distortion about efficiency leads to a significant change in the distribution results in the first step. As we can see from the results of the calculations presented in the Table 3, the information distortion about efficiency does not lead to a change in the distribution results in the tenth step.
Figure 4 presents the averaged values of the relative deviations from the optimal resource values in games with BRM (exact information about the effectiveness of players, distorted information about the effectiveness of players).
600000 0
289962,8 79677,78 89831,48 288575,2 82215,97 90110,22 285095,7 88318,84 90772,57 288250,2 90169,83
Total profit 459472 460901,4 464187,1 461228,1
You can see that difference between total profit in the different experiments consists less of than 1%. We consider this deviation to be insignificant.
The article discusses the effectiveness of distorting information about the agents’ effectiveness in incomplete information conditions in the limited resource distribution problem.
Experiments were performed by robots with various input parameters combinations. A comparative analysis of the games results with reliable and inaccurate information about the players effectiveness was carried out. The agents’ profit and the system total profit are calculated.
The conducted experiments showed that a single distortion of information about the effectiveness of players, with constant distortion of players 'requests, does not affect the distribution of players' profits.