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Appendix 3 - THEORY UNDERLYING THE GAME DESIGNIntroductionGame theory provides the foundation for predicting the decisions of rational agents in strategicsituations. For simple games, it is often possible to find strategic solutions in which no agent canbenefit by changing their strategy (i.e., Nash equilibria). But where the possible strategy space of agame is very large (e.g. if optimal play is contingent upon dynamic local conditions such as resourcedistribution or game history), analytical solutions are often intractable (Hamblin 2013). To ensuresufficient realism and motivations for play, our treatments model many elephants movingindependently and stochastically among spatially explicit landscape cells, and we allow for thedecisions of current rounds to potentially affect payoffs in future rounds (e.g. shooting elephantssubsequently reduces their number). While this critical game realism precludes us from derivinganalytical solutions for optimal play, it is possible to derive analytical solutions for simplifiedconditions (e.g. a single round of game play and expected elephant distribution), and to explore theconsequences of dominant (though not necessarily optimal) strategies (such as “always scare whenelephants are on a cell, else farm”) that might be used in game play.Stakeholders in our games needed to consider the discrete movement of elephants on a spatiallyexplicit landscape, while simultaneously considering how current decisions might affect futurepayoffs. Under such complex conditions, considering the full range of possible strategies available toplayers is not tractable, nor would it be particularly useful for understanding actual stakeholderdecision making in our behavioural games. Nevertheless, it is worthwhile to relate the behaviouralgames being played back to first principles of game theory. In this supplementary material, weanalyse a simplified version of the behavioural game from the main text and demonstrate that whilefarming all landscape cells is a Nash equilibrium, cooperative play to build elephant habitats canultimately lead to higher payoffs if the temptation to defect can be avoided. We also show thepayoffs associated with heuristic strategies played when elephant distributions are discrete acrossthe landscape, and when shooting elephants can have long-term consequences on accrued payoffsin late rounds of the behavioural game. Finally, we show all R code used to analyse Nash equilibria.This supplementary material is organised as follows.1.Nash equilibria for simplified game2.Issues arising from elephant distributions3.Issues arising from sequential rounds4.Supporting code: Annotated functionsIn the first section, we consider a game played for a single round, and given expected (i.e.,probabilistic) rather than realised elephant distributions.Nash equilibria for simplified gameA Nash equilibrium is a stable state of strategies for a game, from which no invading strategy canoutperform the resident strategy, hence any individual player performs best by adopting theresident strategy. Below, we have developed code that allows the user to place three identicalresident strategies on the simulated game landscape for any set of game parameter combinations.The test fitness function then iterates every possible invading strategy and checks its fitness againstthe fitnesses of the resident strategy. It does this by simulating the player in the upper right cornerof the game landscape (note that due to landscape symmetry, choice of landscape quadrantdoes not matter). In the elephant games, players can choose from four possible options for each oftheir nine cells:1.Farm2.Scare

3.Cull4.HabitatEach option is associated with points, a cost, and a weight that affects the cell’s attractiveness togeese (and those of neighbouring cells). There are 4! 262144 possible combinations of farm,scare, cull, and habitat choice on the nine cells. Hence, to test whether or not a resident strategy canbe invaded by a different strategy, we cycle through all 262144 possible land use choicecombinations that could possibly invade the resident strategy . If none of these combinationsresults in a higher payoff than the resident strategy (i.e., if the best invading strategy is the residentstrategy), then we have proved through exhaustive search that the resident strategy is a Nashequilibrium for the chosen game conditions.The key simplifying assumption we make in assessing Nash equilibria is that payoffs are calculatedfrom the expected distributions of elephants (based on landscape cell weights) rather than realiseddistributions of individual elephants. For example, on a landscape in which all cells are being farmedand therefore of equal weight and probability of elephant occurrence , each cell is assumed tohave 18/36 0.5 elephants. Where different land-use decisions are made, expected elephantnumbers are adjusted accordingly by cell weights. This simplification preserves the general structureof the game and allows us to investigate it from first principles using game theory. Allowing insteadfor realised elephant distributions would make calculation of Nash equilibria using our methodintractable, as there are 36"# 1.03 10 # possible ways that 18 elephants can be distributedacross the landscape (although this number could be reduced somewhat by identifying symmetrieson the landscape). It would also likely result in complex strategies, conditional upon realisedelephant distributions; we explore such strategies in the following section.The test fitness function works by iterating through all possible invading strategies and calculatingthe payoff of each. If the background strategy is a Nash equilibrium, then the highest payoff scorewill also be the background strategy. All parameter values are included as arguments, which arelisted in Table 1 of the main text, recreated below. In this simplified game, we assume no elephanthabitat subsidy.Far Farm andm scareFarm andkillElephant habitatYield4440Subsidy000X [2, 4, 6]Crop damage veness–30%80%–Habitat neighbourhoodeffectNoneNoneNone 5 weight to neighbourcellsThe parameter values in the table above are set as default arguments in the function test fitness()function, which is shown below.test fitness - function(land,farm points 4,scare points 4,cull points 4,habitat points 0,

){farm cost 0,scare cost 1,cull cost 2,habitat cost 0,farm weight 10,scare weight 5,cull weight 2,habitat weight 90,bump 5,habitat neigh 1,eleph count 18,damage 2,scare prob 0.8,cull prob 0.3,shoot to kill TRUE,replace living FALSEparameters - c(farm points, scare points, cull points, habitat points,farm cost, scare cost, cull cost, habitat cost,farm weight, scare weight, cull weight, habitat weight,bump, habitat neigh, eleph count, damage,scare prob, cull prob, shoot to kill, replace living);perms - expand.grid( c1 1:4, c2 1:4, c3 1:4, c4 1:4, c5 1:4,c6 1:4, c7 1:4, c8 1:4, c9 1:4);tot perms - dim(perms)[1];fit vector - rep(0, tot perms);time elapsed - proc.time();for( strat in 1:tot perms ){temp l - land;temp l[1,4] - perms[strat,1];temp l[1,5] - perms[strat,2];temp l[1,6] - perms[strat,3];temp l[2,4] - perms[strat,4];temp l[2,5] - perms[strat,5];temp l[2,6] - perms[strat,6];temp l[3,4] - perms[strat,7];temp l[3,5] - perms[strat,8];temp l[3,6] - perms[strat,9];temp l - unlist(temp l);land - matrix(data temp l, nrow 6, ncol 6);land pay - calc payoff(land, parameters);strat fitness - sum(land pay[1:3, 4:6]);fit vector[strat] - strat fitness;time check - proc.time();time print - time check - time elapsed;if(time print[3] 30){pct complete - round(strat / tot perms * 100);print(paste("Progress: ", pct complete, "%", sep ""));time elapsed - proc.time();}

}output - list(strategy perms, fitness fit vector, land land);}return( output );Note that the test fitness function relies on the custom function calc payoff to calculate the payoffof a focal strategy (i.e., the payoff of a focal set of land-use decisions, as played in the upper rightcorner of the landscape), which in turn calls several other custom functions. These custom functionsare explained in detail below, but here it is only important that calc payoff calculates the payoff of afocal invading strategy against a selected resident strategy. The loop in the above cycles throughevery possible invading strategy to calculate all possible payoffs. In the output of test fitness, the listof strategies is returned (strategy), along with the fitness of each strategy (fitness; vector elementscorrespond to rows of strategy), and the original landscape (land).Resident farming strategy: To show that farming on all cells is a Nash equilibrium, it is first necessaryto define a landscape as a six by six matrix in which the background strategy land-use choices arebeing played. For farming, cell land use choice takes a value of 1, so the appropriate land is simply amatrix of 1s.proposed NE - matrix(data 1, nrow 6, ncol 6);## [,1] [,2] [,3] [,4] [,5] [,6]## [1,] 1 1 1 1 1 1## [2,] 1 1 1 1 1 1## [3,] 1 1 1 1 1 1## [4,] 1 1 1 1 1 1## [5,] 1 1 1 1 1 1## [6,] 1 1 1 1 1 1The land matrix is then used in the test fitness function, where all the payoffs of all possibleinvading strategies are compared to the payoffs of the resident strategy.fitness results - test fitness(land proposed NE);These results are then summarised with the following function fitness summary.fitness summary - function(results, background NULL, plot FALSE){if(is.null(background) TRUE){background - matrix(data 1, nrow 3, ncol 3);warning("No resident strategy selected: assuming all farming");}fitness - results fitness;strategy - results strategy;land - results land;fit order - order(fitness, decreasing TRUE);top ten - fit order[1:10];payoff - fitness[top ten];most str - strategy[top ten,];res tabl - cbind(payoff, most str);bgstrat - unlist(t(background)[1:9]);permpos - 1;checkstr - 0;time elapsed - proc.time();while(checkstr 0 & permpos dim(strategy)[1]){

sqrdev - (bgstrat - strategy[permpos,])*(bgstrat - strategy[permpos,]);if( sum(sqrdev) 0 ){checkstr - 1;}else{permpos - permpos 1;}time check - proc.time();time print - time check - time elapsed;if(time print[3] 10){pct complete - round(permpos / dim(strategy)[1] * 100);print(paste("Checked: ", pct complete, "%", sep ""));time elapsed - proc.time();}}}last row - c(fitness[permpos], bgstrat);res tabl - rbind(res tabl, last row);rownames(res tabl) - c("Strategy 1", "Strategy 2", "Strategy 3","Strategy 4", "Strategy 5", "Strategy 6","Strategy 7", "Strategy 8", "Strategy 9","Strategy 10", "Resident Strategy");if(plot TRUE){par(mar c(5, 5, 1, 1), lwd 2);hist(fitness, xlab "Strategy Fitness", ylab "Frequency",main "", cex 1.5, cex.lab 1.5, cex.axis 1.5, col "grey");}return(res tabl);The function fitness summary organises the results from test fitness and generates an ordered listof invading strategies by fitness. If the highest fitness strategy is the resident strategy, then it will bethe first listed in the table and the resident strategy will be a Nash equilibrium. The fitness summaryargument background is for the user to set what the equivalent ‘resident’ strategy looks like for theinvader. The reason that the background strategy is not just assumed to be identical to the otherthree players is because an ‘identical’ strategy might actually rely on symmetry in land orientation –e.g., if everyone farms all their squares except the square in the middle of the board.inv bgd - matrix(data 1, nrow 3, ncol 3);results - fitness summary(results fitness results,background inv bgd);Results for a resident strategy of farming all landscape cells are shown below.payoffc c c c c c c c c1 2 3 4 5 6 7 8 9Strategy 127.000001 1 1 1 1 1 1 1 1Strategy 226.688792 1 1 1 1 1 1 1 1Strategy 326.688791 2 1 1 1 1 1 1 1Strategy 426.688791 1 2 1 1 1 1 1 1

Strategy 526.688791 1 1 2 1 1 1 1 1Strategy 626.688791 1 1 1 2 1 1 1 1Strategy 726.688791 1 1 1 1 2 1 1 1Strategy 826.688791 1 1 1 1 1 2 1 1Strategy 926.688791 1 1 1 1 1 1 2 1Strategy 1026.688791 1 1 1 1 1 1 1 2ResidentStrategy27.000001 1 1 1 1 1 1 1 1The payoff is indicated in the second column, while c1 through c9 refer to the invading strategy’slandscape cells ordered by row as below in table form.## [,1] [,2] [,3]## [1,] 1 2 3## [2,] 4 5 6## [3,] 7 8 9Given that the highest fitness strategy is the resident strategy of farming on all cells, with a totalpayoff of 27, we can say that farming on all cells is a Nash equilibrium strategy; if all neighbours arefarming all of their cells, then the best strategy a focal player can have is to also farm all cells.It is important to note that just because farming on all cells is a Nash equilibrium, this does not meanthat farming on all cells also yields the highest payoff per player. Indeed, we can show using thesame method that a cooperative strategy replacing farming with elephant habitat in each player’scentre-most landscape cell yields a higher payoff for each player. Consider the landscape below, andrecall that 4 indicates the choice of elephant habitat.## [,1] [,2] [,3] [,4] [,5] [,6]## [1,] 1 1 1 1 1 1## [2,] 1 1 1 1 1 1## [3,] 1 1 4 4 1 1## [4,] 1 1 4 4 1 1## [5,] 1 1 1 1 1 1## [6,] 1 1 1 1 1 1The above cooperative resident strategy yields more than 27 points, but is not a Nash equilibrium.To demonstrate this, the below code is run as before.proposed NE coop - matrix(data 1, nrow 6, ncol 6);proposed NE coop[3:4, 3:4] - 4;fitness results coop - test fitness(land proposed NE coop);inv bgd coop - matrix(data 1, nrow 3, ncol 3);inv bgd coop[3, 1] - 4; # Habitat in the lower left cornerresults coop - fitness summary(results fitness results coop,background inv bgd coop);The below table shows the results.

payoffc c c c c c c c c1 2 3 4 5 6 7 8 9Strategy

Appendix 3 - THEORY UNDERLYING THE GAME DESIGN Introduction Game theory provides the foundation for predicting the decisions of rational agents in strategic situations. For simple games, it is often possible to find strategic solutions in which no agent can benefit by changing their strategy (i.e., Nash equilibria). But where the possible strategy space of a game is very large (e.g. if optimal .