# Coalition Game

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## WHAT IS IT?

This is a spin-glass and coupled map lattice combination model for a political decision evolutionary game where each agent (patch) can decide to join one of two coalitions in supporting a measure.

The model incorporates nonlinear dynamics and evolutionary game theory to model stakeholder dynamics in (co)evolutionary cooperative games, so that there is a coadaptive learning dynamics on political resources gathered for joining each coalition.

The coalition also illustrates an example of aggregation in complex adaptive systems, where the coalition is consensus-driven to either support or not a certain political measure being placed to voting.

## HOW IT WORKS

Each patch corresponds to a political agent, that must decide in a political negotiation game to join one of two political consensus groups.

Each agent decides whether to join one or the other political consensus group (coalition) depending on the political resources that the agent expects to obtain from joining the group.

The two groups are labelled as a coalition:

+1 : corresponds to coalition A

-1 : corresponds to coalition B

At each round of the game the agents actively gather information evaluating the political resources that they get from adhering to coalition A or to coalition B.

Thus, if R(A,j,t-1) and R(B,j,t-1) are the j-th political agent's evaluated political resources favoring A and B, respectively, there is a first preliminary assessment of political resource gathering resulting from the update:

RC(A,j,t) = (1 - delta) * (1 - b * R(A,j,t-1) ^ 2) + delta * z(t)

RC(B,j,t) = (1 - delta) * (1 - b * R(B,j,t-1) ^ 2) + delta * z(t)

where RC(.,j,t) corresponds to the agent-specific resources' component dynamical variable.

The quadratic map coupled to a gaussian noise term incorporates a local evaluation that, if positive, is favorable towards an adherence to the corresponding coalition and, if negative, is favorable towards an adherence to the alternative.

The resources component is, then, pondered with respect to the political agent's local network of neighboring agents, this leads to the coupling:

R(A,j,t) = (1 - epsilon(A,j,t)) * RC(A,j,t) + epsilon(A,j,t) * h(A,j,t)

R(B,j,t) = (1 - epsilon(B,j,t)) * RC(B,j,t) + epsilon(B,j,t) * h(B,j,t)

where h(.,j,t) corresponds to the mean resources component of the agents in the political radius "r" of the j-th agent in regards to the corresponding coalition, this means that the agent evaluates the r neighbors' political resources in regards to each coalition, so that a tendency to a local political consensus is introduced.

The coupling parameters are set as:

epsilon(A,j,t) = base-epsilon + fitness(A,t-1)

epsilon(B,j,t) = base-epsilon + fitness(B,t-1)

where fitness(s,t-1) is the proportion of agents in the coalition s (s=A,B) at the end of the previous political negotiations round.

In this form, we have a coupled map lattice (CML) model present at the level of the social evaluation of resources.

Now, for the spin glass section, the coalition rule is given by:

sign(R(A,j,t) - R(B,j,t))

where sign(.) is equal to +1 if the argument is positive and -1 if the argument is negative. Given the correspondences (+1 -> A) and (-1 -> B), it follows that if the agent's evaluated political resources favor coalition A (the difference is positive), then, the agent chooses to join that coalition, while if the agent's evaluated political resources favor coalition B (the difference is negative), then, the agent chooses to join that coalition.

After deciding the coalition for the round, the agents evaluate a new measure that is issued for voting.

The measure is such that it can be supported by coalition A or by coalition B, otherwise it is deemed as a common agreement point for both coalitions.

Each coalition agents will, thus, vote either favorably or unfavorably whether or not it supports the measure. The measure is passed if the votes in favor surpass the votes against.

At each round the voting sense is determined by an equiprobable random trial from the set {-1, 0, +1}, with the following results:

-1 -> B supports the measure but A does not;

0 -> both coalitions support the measure (general political consensus);

+1 -> A supports the measure but B does not.

## HOW TO USE IT

There are four sliders in the model that the user can control:

b : the quadratic map parameter;

r : the mean field radius of the political influence network;

base-epsilon : the mean coupling strength to the political influence network;

delta : the noise coupling for the quadratic map.

Different slider values will lead to different patterns of coalition formation and voting.

## REFERENCES

The current model has been used by the author, for teaching modelling techniques in the context of evolutionary game theory to MsC students at ISCSP - Technical University of Lisbon, and is part of the research project:

Complex Quantum Systems Science and Risk Mathematics

## Comments and Questions

patches-own [ coalition ; +1 = "coalition A", -1 = "coalition B" local-resources_A ; local political resources obtained from joining coalition A local-resources_B ; local political resources obtained from joining coalition B resources-component_A ; component for resources-based choice resources-component_B ; component for resources-based choice epsilon ; coupling to political influence network ] globals [ fitness_A ; proportion of agents in coalition A fitness_B ; proportion of agents in coalition B n-agents ; number of political agents dominant-coalition ; if positive means that the dominant coalition is A if negative it is B support-for-measure ; support for the measure: -1 (B supports but A does not); 0 (both support); +1 (A supports but B does not) decision ; the voting results on the measure ] to setup ca ask patches [ ifelse random-float 1.000 <= 0.5 [ set coalition 1 ] [ set coalition -1 ] ; initially, the coalition configuration is randomly distributed for each political agents set local-resources_A random-float 1.000 ; initial political resources' values for joining coalition A set local-resources_B random-float 1.000 ; initiaal political resources'values for joining coalition B ] set n-agents count patches colorscheme end to go calculate-init-fitness-values ; each game round's initial proportions of agents that belong to A and to B choose-coalition ; coalition choice given the new political resources' evaluation measure-proposal ; a new measure is proposed and voted upon colorscheme do-plot end to calculate-init-fitness-values set fitness_A count patches with [coalition = 1] / n-agents set fitness_B count patches with [coalition = -1] / n-agents end to choose-coalition update-political-resources update-coalitions end to update-political-resources ask patches [ if coalition = 1 [ set epsilon base-epsilon + fitness_A ] if coalition = -1 [ set epsilon base-epsilon + fitness_B ] ] ask patches [ set resources-component_A (1 - delta) * ( 1 - b * local-resources_A ^ 2 ) + delta * random-normal 0 1 ] ask patches [ set resources-component_B (1 - delta) * ( 1 - b * local-resources_B ^ 2 ) + delta * random-normal 0 1 ] ask patches [ set local-resources_A (1 - epsilon) * resources-component_A + epsilon * mean [resources-component_A] of patches in-radius r] ask patches [ set local-resources_B (1 - epsilon) * resources-component_B + epsilon * mean [resources-component_B] of patches in-radius r] end to update-coalitions ask patches [ ifelse ( local-resources_A - local-resources_B ) > 0 [ set coalition 1 ] [ set coalition -1 ] ] set dominant-coalition sum [coalition] of patches end to measure-proposal set support-for-measure -1 + random 3 vote end to vote if support-for-measure = -1 [ set decision count patches with [coalition = -1] - count patches with [coalition = 1] ] if support-for-measure = 0 [ set decision n-agents ] if support-for-measure = 1 [ set decision count patches with [coalition = 1] - count patches with [coalition = -1] ] end to colorscheme ask patches [ if coalition = 1 [set pcolor green] if coalition = -1 [set pcolor red] ] end to do-plot set-current-plot "Dominant Coalition" plot dominant-coalition set-current-plot "Decision" plot decision end

There are 2 versions of this model.

## Attached files

File | Type | Description | Last updated | |
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Coalition Game.png | preview | Preview for 'Coalition Game' | over 10 years ago, by Carlos Pedro S. Gonçalves | Download |

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