classdef gymGame % Defines the outcomes of a T period game with a commitment contract % that imposes penalty p if a_star attendances are not achieved, or % instead with a piece-rate incentive of r per attendance properties % Game properties T; % number of periods in game a_star; % commitment contract attendance threshold p; % commitment contract penalty r; % per attendance reward % Parameters for individuals taking up the contract lambda; % 1/mean of exponential cost distribution min_cost; % lower bound on support of cost distribution beta; % actual present focus betatilde; % perceived present focus b; % health benefit mu; % fraction of people taking up the contract % Outcomes under commitment contract utilities; % utilities in each period for each g t1_utility; % period 1 utility utilities_tilde; % perceived utilities in each future period att; % distribution of attendances in simulation avg_att; % average attendance in simulation avg_costs; % average aggregate costs incurred avg_t1_att; % average first period attendance prob_success; % likelihood of meeting threshold in simulation prob_att_c_b; % likelihood of attending when cost > benefit prob_att_day; % likelihood of attending each day % Outcomes under piece-rate incentive(s) prob_att_r; % probability of attending on a given day att_r; % expected attendance under incentive r change_att_r; % expected increase in attendance due to r benefits_r; % expected health benefits under incentive r costs_r; % expected costs under incentive r net_surplus_r; % surplus under incentive r, net of incentive surplus_r; % total surplus under incentive r % Identifier spec; end properties (Constant) seed = 12345; % seed for simulation rounds = 10000; % number of iterations of the simulation end methods function obj = gymGame(properties, parameters, min_cost, title) % input properties obj.T = properties.T; obj.a_star = properties.a_star; obj.p = properties.p; % set parameters from data obj.lambda = parameters.lambda; obj.min_cost = min_cost; obj.beta = parameters.beta; obj.betatilde = parameters.betatilde; obj.b = parameters.b; obj.mu = parameters.mu; % set identifier obj.spec = title; % compute matrix of utilities obj = obj.compute_utilities(); % simulate behavior of participants obj = obj.simulate_behavior(); end function obj = compute_utilities(obj) V_n = NaN([obj.a_star+1 obj.T]); V_tilde = NaN([obj.a_star+1 obj.T]); % set last period utilities for g = 0:obj.a_star g1 = g + 1; % 1-indexed V_n(g1,obj.T) = obj.compute_T_utility(g, 'actual'); V_tilde(g1,obj.T) = obj.compute_T_utility(g, 'perceived'); end % set all other period utilities for period = 1:obj.T-1 t = obj.T - period; % go backwards start = max(obj.a_star - t + 1, 0); % iterate through remaining attendances for g = start:obj.a_star g1 = g + 1; % 1-indexed % expected utility difference from attending delta_V_tilde = V_tilde(max(g1-1,1),t+1) - V_tilde(g1,t+1); delta_V_n = V_n(max(g1-1,1),t+1) - V_n(g1,t+1); % perceived expected utility at period t V_tilde(g1,t) = (obj.F(obj.betatilde.*(obj.b + delta_V_tilde)).* ... (obj.b + delta_V_tilde)) + V_tilde(g1,t+1) - ... obj.C(obj.betatilde.*(obj.b + delta_V_tilde)); % actual expected utility at period t V_n(g1,t) = (obj.F(obj.beta.*(obj.b + delta_V_tilde)).* ... (obj.b + delta_V_n)) + V_n(g1,t+1) - ... obj.C(obj.beta.*(obj.b + delta_V_tilde)); end end obj.utilities = V_n; obj.t1_utility = V_n(obj.a_star+1,1); % period 1 utility obj.utilities_tilde = V_tilde; end function V_T = compute_T_utility(obj, g, focus) % compute utility in the last period % set present focus parameter if strcmp(focus, 'actual') B = obj.beta; elseif strcmp(focus, 'perceived') B = obj.betatilde; end if g == 0 % no attendances left V_T = (obj.F(B.*obj.b).*obj.b) - obj.C(B.*obj.b); elseif g == 1 % one attendance left V_T = (obj.F(B.*(obj.b + obj.p)).*obj.b) - ... obj.C(B.*(obj.b + obj.p)) - ... ((1 - obj.F(B.*(obj.b + obj.p))).*obj.p); else % more than one attendance left V_T = (obj.F(B.*obj.b).*obj.b) - obj.C(B.*obj.b) - obj.p; end end function obj = simulate_behavior(obj) % simulate behavior through T periods for a number of rounds to % generate estimates of average attendance and the likelihood % of meeting the contract threshold rng(obj.seed); % set seed obj.att = zeros(1,obj.rounds); % default no attendances t1_att = obj.att; att_day = zeros(obj.T,obj.rounds); % default no attendances costs_day = zeros(obj.T,obj.rounds); % default no costs met_goal = zeros(1,obj.rounds); % default failed to meet goal obj.prob_att_c_b = zeros(1,obj.rounds); % default never attend for round = 1:obj.rounds c = obj.cost_draws(); % generate cost draws g = obj.a_star; % all attendances initially left a = 0; % total number of attendances c_b = 0; % events where cost > benefit att_c_b = 0; % attendance in those events % check if the DM wants to go to the gym in each period for t = 1:obj.T g1 = g + 1; % 1-indexed if t == obj.T && (g == 0 || g >= 2) % last period, case i if c(t) <= (obj.beta * obj.b) % update attendances & costs incurred g = max(g - 1, 0); a = a + 1; att_day(t,round) = 1; costs_day(t,round) = c(t); if c(t) > obj.b % cost is greater than benefit att_c_b = att_c_b + 1; end end elseif t == obj.T && g == 1 % last period, case ii if c(t) <= (obj.beta * (obj.b + obj.p)) % update attendances & costs incurred g = max(g - 1, 0); a = a + 1; att_day(t,round) = 1; costs_day(t,round) = c(t); if c(t) > obj.b % cost is greater than benefit att_c_b = att_c_b + 1; end end else % all other periods if c(t) <= (obj.beta * (obj.b + ... obj.utilities_tilde(max(g1-1,1),t+1) - ... obj.utilities_tilde(g1,t+1))) % update attendances & costs incurred g = max(g - 1, 0); a = a + 1; att_day(t,round) = 1; costs_day(t,round) = c(t); if c(t) > obj.b % cost is greater than benefit att_c_b = att_c_b + 1; end end end if c(t) > obj.b % cost is greater than benefit c_b = c_b + 1; end if t == 1 && g < obj.a_star % period one attendances t1_att(round) = 1; end end if g == 0 met_goal(round) = 1; % input outcome end obj.att(round) = a; % input attendances obj.prob_att_c_b(round) = att_c_b ./ c_b; end % return means obj.avg_att = mean(obj.att); obj.avg_t1_att = mean(t1_att); obj.prob_success = mean(met_goal); obj.prob_att_c_b = mean(obj.prob_att_c_b); obj.prob_att_day = sum(att_day,2) ./ obj.rounds; obj.avg_costs = mean(sum(costs_day,1)); end function obj = compute_incentive_behavior(obj, r) % compute expected attendance and total surplus given % piece-rate incentive r obj.r = r; % compute expected attendance on a given day obj.prob_att_r = obj.F(obj.beta .* (obj.b + r)); prob_att_no_r = obj.F(obj.beta .* (obj.b)); % without incentive % compute expected attendance obj.att_r = obj.T .* obj.prob_att_r; % compute expected increase in attendance obj.change_att_r = obj.T .* (obj.prob_att_r - prob_att_no_r); % compute expected benefits obj.benefits_r = obj.att_r .* obj.b; % compute expected costs obj.costs_r = obj.T .* obj.C(obj.beta .* (obj.b + r)); % compute surplus net of incentive obj.net_surplus_r = obj.benefits_r - obj.costs_r; % compute total surplus obj.surplus_r = obj.net_surplus_r + (obj.att_r .* r); end function probability = F(obj, x) % cumulative distribution function of costs z = x - obj.min_cost; z = gymGame.minzero(z); probability = 1 - exp(-obj.lambda.*z); end function density = f(obj, x) % cost density function % returns zero when given x < lower bound on costs z = x - obj.min_cost; density = obj.lambda.*exp(-obj.lambda.*z); density = density.*heaviside(heaviside(z)); end function expectation = C(obj, x) % expected cost function z = x - obj.min_cost; z = gymGame.minzero(z); expectation = (1 ./ obj.lambda).*(1 - exp(-obj.lambda.*z)) ... - (z.*exp(-obj.lambda.*z)) + obj.min_cost.*obj.F(x); end function c = cost_draws(obj) % vector of random cost draws for each period mean = 1 ./ obj.lambda; c = exprnd(mean, [1 obj.T]) + obj.min_cost; end end methods(Static) function y = minzero(x) % returns x if x is positive, 0 otherwise y = x .* heaviside(heaviside(x)); end end end