% optimize restaurant strategy at equilibrium
% hold menu price, R, total people constant
% hold competition parameters constant
% select tip rate T1, and base pays bw1, bc1 (within legal limits) to maximize profit
function Tc = compute_Tc(m1,m2,T1,T2,R,bw1,bw2,bc1,bc2,min_bw,min_bc,Nd,Nw,Nc,t0,tend,IC)
tip_conventional = 0:0.05:0.5;
tip_conventional = [0,0.01,0.02,0.03,0.04,0.05,0.1,0.15,0.2,0.25,0.3,0.4,0.5];
% maximize profit for range of tip rates
[pmax1,~,~] = max_profit2(t0,tend,m1,m2,tip_conventional,tip_conventional,R,bw2,bc2,min_bw,min_bc,Nd,Nw,Nc,IC,[bw1,bc1]);
% maximize profit for no tipping allowed (T2 is increasing vector of conventional tip rates)
[pmax2,~,~] = max_profit2(t0,tend,m1,m2,zeros(1,length(tip_conventional)),tip_conventional,R,bw2,bc2,min_bw,min_bc,Nd,Nw,Nc,IC,[bw1,bc1]);

% residuals between strategies
res = pmax1-pmax2; res_sign = res(1:length(res)-1).*res(2:end);
% index before strategy switch
ind = find(res_sign<=0);
% check if Tc = 0 is not only switch
if ind(end)>1
    % index of first switch after Tc = 0 (all numerical exploration indicates there are at most 2)
    ii = ind(2);
    % (x1,y1) and (x2,y2) are line segment for pmax1
    x1 = tip_conventional(ii); y1 = pmax1(ii); x2 = tip_conventional(ii+1); y2 = pmax1(ii+1);
    % (x3,y3) and (x4,y4) are line segment for pmax2
    x3 = x1; y3 = pmax2(ii); x4 = x2; y4 = pmax2(ii+1);
    % find x value of intersection (https://en.wikipedia.org/wiki/Line%E2%80%93line_intersection#Given_two_points_on_each_line)
    Tc = ((x1*y2-y1*x2)*(x3-x4)-(x1-x2)*(x3*y4-y3*x4))/((x1-x2)*(y3-y4)-(y1-y2)*(x3-x4));
else
    Tc = 0;
end

% figure
% plot(tip_conventional,pmax1,'o-k')
% hold on
% plot(tip_conventional,pmax2,'o-r')
% line([Tc Tc], [min([pmax1; pmax2]) max([pmax1; pmax2])]);
% xlabel('conventional tip rate','FontSize',14)
% ylabel('restaurant profitibility (unitless)','FontSize',14)
% legend('allow tipping','forbid tipping')

end