mdp_eval_policy_optimality(P, R, discount, Vpolicy)

function [is_multiple, optimal_actions] = mdp_eval_policy_optimality(P, R, discount, Vpolicy)

% mdp_eval_policy_optimality   Eval if near optimum actions exists for each state

% Arguments -------------------------------------------------------------
% Let S = number of states, A = number of actions
%   P(SxSxA)  = transition matrix
%              P could be an array with 3 dimensions or 
%              a cell array (1xA), each cell containing a matrix (SxS) possibly sparse
%   R(SxSxA) or (SxA) = reward matrix
%              R could be an array with 3 dimensions (SxSxA) or 
%              a cell array (1xA), each cell containing a sparse matrix (SxS) or
%              a 2D array(SxA) possibly sparse  
%   discount  = discount rate in ]0; 1[
%   V(S)      = optimum value function 
% Evaluation -------------------------------------------------------------
%   is_multiple  = true when at least a state has several near optimal actions, false if not
%   optimal_actions(SxS) = boolean matrix, optimal_actions(s,s') is true when
%                          Q(s,s') - Vpolicy(s) < 0.01 else false

% MDPtoolbox: Markov Decision Processes Toolbox
% Copyright (C) 2009  INRA
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%      this list of conditions and the following disclaimer in the documentation 
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%    * Neither the name of the <ORGANIZATION> nor the names of its contributors 
%      may be used to endorse or promote products derived from this software 
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% check of arguments
if discount <= 0 | discount >= 1
    disp('--------------------------------------------------------')
    disp('MDP Toolbox ERROR: Discount rate must be in ]0; 1[')
    disp('--------------------------------------------------------')
elseif (iscell(P)) & (size(Vpolicy) ~= size(P{1},1))
    disp('--------------------------------------------------------')
    disp('MDP Toolbox ERROR: Vopt must have the same dimension as P')
    disp('--------------------------------------------------------')
elseif (~iscell(P)) & (size(Vpolicy) ~= size(P,1))
    disp('--------------------------------------------------------')
    disp('MDP Toolbox ERROR: Vpolicy must have the same dimension as P')
    disp('--------------------------------------------------------') 
else    
    
    % compute Q(SxA)
    PR = mdp_computePR(P,R);
    if iscell(P)
        S = size(P{1},1);
        A = length(P);
        for a=1:A           
            Q(:,a)= PR(:,a) + discount*P{a}*Vpolicy;
        end
    else
        S = size(P,1);
        A = size(P,3);
        for a=1:A
            Q(:,a)= PR(:,a) + discount*P(:,:,a)*Vpolicy;
        end
    end
        
    % search near optimal actions a for each state s, satisfaying
    % Q(s,a) - Q(s, a*) < epsilon
    % where a* is the optimal action for state s
    epsilon = 0.01;
    optimal_actions = (abs(Q - repmat(Vpolicy,1,A))< epsilon);

    if max(sum(optimal_actions,2)) == 1
        is_multiple = false;
    else 
        is_multiple = true;
    end;
end