simulation of barter processes

8/8/2019 Simulation of Barter Processes

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Simulation of barter processes as

heuristic method for multiobjectiveoptimization

Stefano Nasini


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1) Bargaining problem2) The associated Game

3) Simulation model with Petri Net4) Java implementation5) Experimental design

6) Conclusions and Results


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The k -lateral bargaining problem concerns the ways k agents cooperate when non-cooperation leads to Pareto-inefficient output in the allocation of a fixed quantity. Such aproblem has been modelled from different outlooks and still represents a source of uncertainty in Social Science and Decision Theory.

Bargaining problem

Suppose two agents are endowed with tworespective points in the 3-dimensional commodity

space and utility function which represents their preference relation has been defined.

The related bargaining problem consists in findingtwo new points which keep untouched the overall amount of endowments.

0

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),,(max

),,(max

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x x xu


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Bargaining problem

As far as the determination of a unique solution isconcerned, Nash suggested that a solution shouldfulfil certain conditions:

-symmetry,- invariant to affine transformations,- Pareto efficiency,- independence of irrelevant alternatives.

The solution proposed by Nash considers the compact and convex payoff space P = {(u1, u2)|u1 > t1; u1 > t2}, where (t1, t2) is the vector of disagreement payoff, and a set H of non-dominatedvectors of P.


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The associated Game

Consider a lexicographic order in the Cartesian product of the commodities type. Let (i,j) be an elementof such a set of ordered pares. Then, a sequential game is defined as follows. The first element (i,j) isselected and an intermediary chooses within three actions:

])',(),',([21 ji ji x xUtility y yUtility


])(),,([ ,21 ji ji x xUtility y yUtility

otherwise0

bydominatedParetot isn'if 1payoff(

)],(),,( 21[ ji ji y yUtility x xUtility)

The payoff function of the intermediary is the following


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The associated Game

The intermediary proposes exactly that tradewhich isnt Pareto-dominated by

. If the thirdstrategy has been chosen, then the secondcouple of good (i,j) is selected and theintermediary must chosen again between

and .





Then the two agent sequentially choosewhether to accept the proposal, inaccordance with their associated utilityfunctions.


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Simulation model with Petri Net

We used a ColoredTimestamp Petri Net. Eachtoken has a timestamp,which is a particular colorused to specify the earliest time when it can be

consumed.

The transition consumes @units firing time.

In an economy with three commodities there will be 4 tokens in p.1 at initial marking. The first transaction to fire is t.2 because it is the unique with @=0. Then, the clock will be updated as@++ so that t.1 will fire since the timestamps associated to its incoming tokens are @ = 1,which is the value of the current clock. The clock is again updated (clock = 2) and the next

transaction to fire will be t.1 .


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Uti_A =all amount;@ ++;

true;i = 1;@ ++;Iterations++;

if ( Utility(u_a, All_A+y-z) > Uti_A && Utility(u_b, All_B-y+z) > Uti_A ){

Uti_A = Utility(u_a, All_A+y-z);Uti_B = Utility(u_b, All_B-y+z);All_A = All_A+y-z;All_B = All_B-y+z;arc.1 = (Uti_A, @+1);

arc.2 = (Uti_B, @+1);arc.3 = 2y;arc.4 = 2z;

if ( M(p.6, t) < 3) { arc.6 = (1, @+3) } else { arc.6 = (0, @+3) }}if ( Utility(u_a, All_A-y+z) > Uti_A && Utility(u_b, All_B+y-z) > Uti_A ){

Uti_A = Utility(u_a, All_A-y+z);Uti_B = Utility(u_b, All_B+y-z);All_A = All_A-y+z;All_B = All_B+y-z;arc.1 = (Uti_A, @+1);arc.2 = (Uti_B, @+1);arc.3 = 2z;arc.4 = 2y;if ( M(p.6, t) < 3) { arc.6 = (1, @+3) } else { arc.6 = (0, @+3) }

}else {

arc.1 = Uti_A;arc.2 = Uti_B;arc.3 = y + z;arc.4 = y + z;arc.6 = (-1, @+3);

}

@ ++;

Simulation model with Petri Net


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Simulation model with Petri NetThe sequence beside showthe reachability tree of thePetri Net previously shown.

We consider a three-commodities two - agentseconomy with the followinginitial condition.

A given marking isreachable from the initialmarking if exists asequence of firings that transform the initialmarking into it.

1]6,[8,=

11]7,[5,=

50]110,[150,=

140]250,[100,=

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Java implementation

We implemented the simulation model in Java, a programming languageoriginally developed by James Gosling at Sun Microsystems. There werefive primary goals in the creation of the Java language:

- It should be "simple, object oriented, and familiar".- It should be "robust and secure".

- It should be "architecture neutral and portable".- It should execute with "high performance"

We use a Netbeans Integrated Development Environment to interpret the Java codeand run it in Windows Environment. An Integrated Development Environment is a

software application that provides comprehensive facilities for software developmentand usually consists of:

- a source code editor;- a compiler and/or an interpreter;- build automation tools;

- a debugger;


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The NetBeans Integrated Development

Environment is written in Java and runseverywhere with a JVM.

We implemented two java classes:

Java implementation

- public class RepeatedGame { . . . }- public static void main(String[] args) { . . . }

The result shown beside allow a quick understanding of the improvement theplayers get after the trading are carried out.


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Experimental design

X1: Correlation between utility and quantity of the same agent :+ : mean correlation higher then 0.7;: mean correlation within [ -0.4 , 0.4];-: mean correlation lower then -0.7 .

X2> Correlation between utility and quantity of the opposite agents:+ : mean correlation higher then 07.;: mean correlation within [ -0.4 , 0.4];

-: mean correlation lower then -0.7 .

We use a two-factors-three-levels experiment (3 2 factorial designs) with four replications foreach combination of them to check whether the number of trades needed to reach a Paretoefficient allocation depends on the correlation between utility and quantity of the initialendowment.

)N(0,~ 2

211222110

k

k k k k k k x x x x y

A factorial design of experiment consists of two or more factors,each with discrete

possible values or "levels", and whoseexperimental unitstake on all possiblecombinations of theselevels across all such

factors.


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X1 X2 Responces

+ + 45 41 36 31

+ 35 5 6 4

+ - 33 3 1 2

+ 25 48 5 71

1 21 1 20

- 11 8 1 2

- + 69 82 198 103

- 43 21 36 51

- - 2 20 5 20

Data are collected in the following table where each row has 4 observations of the number of trades needed to reach a Pareto efficient allocation for each combination of factors. The idea isthat there are many possible initial allocations which can fulfill the definition of a factorcombination (i.e. many integer allocations could have mean correlation higher then 0.7 betweenutility and quantity of the same agent and mean correlation higher then 0.7 between utility andquantity of opposite agents). Among then we randomly select fours, which constitute fourdistinct replications for the same combination of factors.

Experimental design

The number of replications has beencalculated as follwos


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Experimental design

Effect Estimate Stand.Err t.value p.value

intercept 30.444 4.355 6.991 6.39e-08 ***

X1 - 16.583 5.334 - 3.109 0.00392 **

X2 27.333 5.335 5.125 1.38e-05 ***

X1*X2 - 18.812 6.533 - 2.880 0.00704 **

By maximum likelihood,we obtain the followingestimation of the effectsand the associatedp.values.

The following two plots suggest the presence of a quadratic effect of X1 (Correlation between

utility and quantity of the same agent ). Instead, the evvect of X2 over the responce seems to belinear. Nonetheless, ti could be reasonable to think that such a curvature could be explained bythe interaction between X1 and X2.


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Conclusions and Results

-The correlation between utility and quantity of the same agent is affecting the number of movements of the Local Search needed to reach a Pareto efficient allocation.

- Also the correlation between utility and quantity of opposite agents has a significativeeffect on the response. The underlying idea is that the more preferences of agents matchtheir respective allocations the closer they are to a Pareto efficient allocation. Conversely,the more preference of an agent matches with the allocation to the opposite agent themore it is likely that they have both a convenience in carrying out a movement.

- We also observe something that is less intuitive; that is, the effect of both factors on theresponse is not additive: there exists a signifivative interaction between the factors.

simulation of barter processes

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