spatial prisoner's dilemma

[pedro-andrade-inpe]

Finish the spatial prisoner's dilemma model in TerraME. It needs to deal with empty cells. The first version is shown below.

--[[

Games on Grids
Martin A. Nowak and Karl Sigmund

Spatial evolutionary game theory, which was first used to shed light on the emergence of
cooperation, has grown rapidly during the past five years and has proved useful in other
biological and economic contexts. 

A wealth of computer simulations of artificial populations now exist showing
that cooperation can be stably sustained in societies of simple automata.
The introduction of spatial structure has shown that cooperation becomes
even more likely if interactions are restricted to neighbors. The venerable
rule of cooperating with neighbors is certainly not new. But placed alongside
other spatial models, it offers wide perspectives for the emergence and
stability of cooperative societies. In particular, it shows that even if the interaction
between two individuals is not repeated, cooperation can be sustained
in the long run.

Spatial Prisioner's Dillema

Under what conditions will there be a spatial coexistence of cooperators and defectors, and
When will we observe the extinction of either cooperators or defectors?

What does the variety and the size of the spatial domains depend on?

Will the dynamics always lead to stationary patterns, or will there be also non-stationary
patterns in the long run?

How does the dynamics changes if a larger local neighborhood is considered in the update
rule (i.e. if the second nearest neighbors are included, too)?

The following example uses the following rules:

Cooperators can survive by forming clusters and thereby outweighing losses against defectors.
There is a balance between payoff and survival rates

The following example uses the following rules:

1. Each player occupies a single cell
2. Players compete against all 8 neighbors in an 3 x 3 cell 
3. The initial configuration has one defector in the center
4. Updates are synchronous 
5. Each cell copies the most successful strategy in its neighborhood

References 

Nowak MA & May RM (1992). Evolutionary games and spatial chaos. 
        Nature 359:826?~@~S829

Schweitzer et al. (2002): "Evolution of Cooperation in a Spatial Prisoner?~@~Ys Dilemma"
        Advances in Complex Systems, vol. 5, no. 2-3, pp. 269-299

Hauert, Ch. & Doebeli, M. (2004) Spatial structure often inhibits the evolution of cooperation
        in the spatial Snowdrift game. Nature (2004) 428 643-646.

]]--

--[[ 
1. Payoffs
   (T)emptation, (R)eward, (P)unishment, and (S)ucker's payoff
   T > R > P > S  -- prisioner's dillema
   T > R > S > P  -- hawk-dove game

-- Nowak and May models
    3 x 3 ("moore") neighborhood, 
    agent is a neighbor of itself
    Prisioner's dillema with R = 1.00 and S = 0.00

-- Schweitzer models
    2x2 ("vonneumann") neighborhood, 
    agent is not a neighbor of itself
    Prisioner's dillema with R = 3.00 and S = 0.00

2. Regions on Novak May models
    1.75 < T < 1.8  -- cooperators prevail over a network of defectors
    1.80 < T < 2.0  -- spatial chaos

    if T = 1.85, put one defector in the center cell - spatial chaos 


3. Regions on Schweitzer models

-- Consider R = 3 and S = 0 
-- Vary 3 < T < 6 and 0 < P < 3
-- T > R > P > S (prisioners dillema)

-- Region A Coexistence with a majority of cooperators - 
--     3 < T < (4 ?~H~R P/3)  if 0.00 < P < 0.75 
--     3 < T < (6 ?~H~R 3P)   if 0.75 < P < 1.0
--    examples: (T = 3.5, P = 0.5 ) and (T = 3.8, P = 0.6)

-- Region B - Coexistence with a minority of cooperators - spatial chaos
--     4 ?~H~R P/3 < T  < 4.5?~H~RP  if  0.0 < P < 0.75
--    examples: (T = 3.9, P = 0.5) and (T = 4.2, P = 0.2)

-- -- limit btw A and B - T = 3.8333 and P = 0.5 

-- Region C Coexistence with cooperators in small clusters (region C)
--
--   4 - P/3 < T < 9 - 3P iff 1.5 < P < 1.875
--   4 - P/3 < T < 6 - P  iff 0.75 < P < 1.5
--   4.5 - P < T < 6 - P  iff 0.0 < P < 0.75
--   examples: (T = 4.5, P = 1.0) and (T = 5.6, P = 0.2)

--  (T=4.0, P =0.5) is the border between regions B and C.
]]--

payoffs = {
    nowak_may = {
        fractal = {temptation = 1.85, reward = 1.00, punishment = 0.01, sucker = 0.00},
        coop1   = {temptation = 1.45, reward = 1.00, punishment = 0.01, sucker = 0.00},
        coop2   = {temptation = 1.76, reward = 1.00, punishment = 0.01, sucker = 0.00},
        chaos   = {temptation = 1.90, reward = 1.00, punishment = 0.01, sucker = 0.00}
    },
    schweitzer = {
        regA     = {temptation = 3.50, reward = 3.00, punishment = 0.50, sucker = 0.00},
        regB     = {temptation = 3.90, reward = 3.00, punishment = 0.50, sucker = 0.00},
        regC     = {temptation = 4.50, reward = 3.00, punishment = 1.00, sucker = 0.00},
        borderAB = {temptation = 3.83, reward = 3.00, punishment = 0.50, sucker = 0.00},
        borderBC = {temptation = 4.00, reward = 3.00, punishment = 0.50, sucker = 0.00}
    }
}

percent_cooperators = {
    nowak_may  = {fractal = 1.00, coop1 = 0.90, coop2  = 0.90, chaos  = 0.90},
    schweitzer = {regA    = 0.50, regB  = 0.75, regC   = 0.90, borderAB = 0.50, borderBC = 0.50}
}

local probCooperate = 0

-- create a table with the available strategies in order to
-- allow the user to select one from the graphical interface
strategies = {}
idxstrategies = {}

forEachOrderedElement(percent_cooperators, function(idx, mtable)
    forEachOrderedElement(mtable, function(midx)
        local value = idx.."_"..midx
        table.insert(strategies, value)
        idxstrategies[value] = {author = idx, case = midx}
    end)
end)

-- neighbor configuration
neigh = {
    nowak_may  = {strategy = "moore", self = true, wrap = true},
    schweitzer = {strategy = "vonneumann", self = false, wrap = true}
}

-- result of a game between two agents
local game = {}
game.Cooperate = {}
game.Defect    = {}

-- state of an agent comparing previous and current move
state = {}
state.Cooperate = {}
state.Defect    = {}
state.Cooperate.Cooperate =  "Cooperate"
state.Cooperate.Defect    =  "CooperateToDefect"
state.Defect.Cooperate    =  "DefectToCooperate"
state.Defect.Defect       =  "Defect"

local function playGame(agent1, agent2)
    return game[agent1.strategy][agent2.strategy]
end

local agent = Agent{
    playWithNeighbors = function(agent)
        local cell = agent:getCell()
        agent.payoff = 0

        forEachNeighbor(cell, function(cell, neigh)
            agent.payoff = agent.payoff + playGame(agent, neigh:getAgent())
        end)
    end,
    findBestStrategy = function(agent)
        local cell = agent:getCell()
        agent.beststrategy = agent.strategy
        local bestpayoff = agent.payoff

        forEachNeighbor(cell, function(cell, neigh)
            other = neigh:getAgent()
            if other.payoff > bestpayoff then
                bestpayoff   = other.payoff
                agent.beststrategy = other.strategy
            end
        end)
    end,
    changeStrategy = function(agent)
        agent.state    = state[agent.strategy][agent.beststrategy]
        agent.strategy = agent.beststrategy
    end,
    init = function(self)
        if math.random() <= probCooperate then
            self.strategy = "Cooperate"
        else
            self.strategy = "Defect"
        end

        self.state        = self.strategy
        self.beststrategy = self.strategy
    end
}

function defectorCentre(model)
    -- find the central cell
    local mid = (model.dim - 1) / 2
    local cell = model.cells:get(mid, mid)
    -- get the agent at the central cell
    local ag = cell:getAgent()
    ag.strategy = "Defect"
    ag.state    = "Defect"
end

SPD = Model{
    dim = Choice{min = 10, default = 41},
    agents = Choice{min = 100, default = 1681},
    finalTime = Choice{min = 50, default = 100},
    strategy = Choice(strategies),
    init = function(model)
        verify(model.agents <= model.dim * model.dim, "There should be enough space for all agents.")

        local author = idxstrategies[model.strategy].author
        local case = idxstrategies[model.strategy].case

        game.Cooperate.Cooperate = payoffs[author][case].reward
        game.Cooperate.Defect    = payoffs[author][case].sucker
        game.Defect.Cooperate    = payoffs[author][case].temptation
        game.Defect.Defect       = payoffs[author][case].punishment

        probCooperate = percent_cooperators[author][case]

        model.cell = Cell{
            color = function(cell)
                local agent = cell:getAgent()

                if agent then
                    return agent.state
                else
                    return "Empty"
                end
            end
        }

        model.cells = CellularSpace{
            xdim = model.dim,
            ydim = model.dim,
            instance = model.cell
        }

        model.cells:createNeighborhood(neigh[author])

        model.society = Society{
            quantity = model.agents,
            instance = agent,
            percentCooperators = function(self)
                local num_coop = 0

                forEachAgent(self, function(agent)
                    if agent.strategy == "Cooperate" then
                        num_coop = num_coop + 1
                    end
                end)

                return (100 * num_coop) / #self
            end
        }

        model.env = Environment{model.cells, model.society}

        model.env:createPlacement{
            strategy = "uniform"
        }

        if author == "nowak_may" and case == "fractal" then
            defectorCentre(model)
        end

        model.timer = Timer{
            Event{action = function()
                model.society:playWithNeighbors()
                model.society:findBestStrategy()
                model.society:changeStrategy()

                model.society:notify()
                model.cells:notify()
            end}
        }

        Map{
            target = model.cells,
            select = "color",
            color = "RdBu",
            value = {"Defect", "CooperateToDefect", "Empty", "DefectToCooperate", "Cooperate"}
        }

        Chart{
            target = model.society,
            select = "percentCooperators"
        }

        model.society:notify()
    end
}

SPD:configure()