// Given square matrix of Prob[i>j] returns a column vector for Prob[i].
// Uses Markov process, not 1-step conditional probability.
// Challenges have uniform probability 1/N
KMatrix Model::markovUniformPCE(const KMatrix & pv) {
const double pTol = 1E-6;
unsigned int numOpt = pv.numR();
auto p = KMatrix(numOpt, 1, 1.0) / numOpt; // all 1/n
auto q = p;
unsigned int iMax = 1000; // 10-30 is typical
unsigned int iter = 0;
double change = 1.0;
while (pTol < change) {
change = 0;
for (unsigned int i = 0; i < numOpt; i++) {
double pi = 0.0;
for (unsigned int j = 0; j < numOpt; j++) {
pi = pi + pv(i, j)*(p(i, 0) + p(j, 0));
}
assert(0 <= pi); // double-check
q(i, 0) = pi / numOpt;
double c = fabs(q(i, 0) - p(i, 0));
change = (c > change) ? c : change;
}
// Newton method improves convergence.
p = (p + q) / 2.0;
iter++;
assert(fabs(sum(p) - 1.0) < pTol); // double-check
}
assert(iter < iMax); // no way to recover
return p;
}
void RP2Model::setRP2(const KMatrix & pm0) {
const unsigned int nr = pm0.numR();
const unsigned int nc = pm0.numC();
if (0 < numAct) {
assert(nr == numAct);
}
else {
numAct = nr;
}
if (0 < numOptions()) {
assert(nc == numOptions());
}
else {
theta.resize(nc); // was size zero
for (unsigned int i = 0; i < nc; i++) {
theta[i] = i;
}
}
assert(minNumActor <= numAct);
assert(numAct <= maxNumActor);
assert(minNumOptions <= numOptions());
for (auto u : pm0) {
assert(0.0 <= u);
assert(u <= 1.0);
}
// if all OK, set it
polUtilMat = pm0;
return;
}
KMatrix rescaleRows(const KMatrix& m1, const double vMin, const double vMax) {
assert(vMin < vMax);
const unsigned int nr = m1.numR();
const unsigned int nc = m1.numC();
KMatrix m2 = KMatrix(nr, nc);
for (unsigned int i = 0; i < nr; i++) {
double rowMin = m1(i, 0);
double rowMax = m1(i, 0);
for (unsigned int j = 0; j < nc; j++) {
const double mij = m1(i, j);
if (mij < rowMin) {
rowMin = mij;
}
if (mij > rowMax) {
rowMax = mij;
}
}
const double rowRange = rowMax - rowMin;
assert(0 < rowRange);
for (unsigned int j = 0; j < nc; j++) {
const double mij = m1(i, j);
const double nij = (mij - rowMin) / rowRange; // normalize into [0, 1]
const double rij = vMin + (vMax - vMin)*nij; // rescale into [vMin, vMax]
m2(i, j) = rij;
}
}
return m2;
}
KMatrix Model::bigRfromProb(const KMatrix & p, BigRRange rr) {
double pMin = 1.0;
double pMax = 0.0;
for (double pi : p) {
assert(0.0 <= pi);
pMin = (pi < pMin) ? pi : pMin;
pMax = (pi > pMax) ? pi : pMax;
}
const double pTol = 1E-8;
assert(fabs(1 - KBase::sum(p)) < pTol);
function<double(unsigned int, unsigned int)> rfn = nullptr;
switch (rr) {
case BigRRange::Min:
rfn = [pMin, pMax, p](unsigned int i, unsigned int j) {
return (p(i, j) - pMin) / (pMax - pMin);
};
break;
case BigRRange::Mid:
rfn = [pMin, pMax, p](unsigned int i, unsigned int j) {
return (3 * p(i, j) - (pMax + 2 * pMin)) / (2 * (pMax - pMin));
};
break;
case BigRRange::Max:
rfn = [pMin, pMax, p](unsigned int i, unsigned int j) {
return (2 * p(i, j) - (pMax + pMin)) / (pMax - pMin);
};
break;
}
auto rMat = KMatrix::map(rfn, p.numR(), p.numC());
return rMat;
}
// returns a square matrix of prob(OptI > OptJ)
// these are assumed to be unique options.
// w is a [1,actor] row-vector of actor strengths, u is [act,option] utilities.
KMatrix Model::vProb(VotingRule vr, VPModel vpm, const KMatrix & w, const KMatrix & u) {
// u_ij is utility to actor i of the position advocated by actor j
unsigned int numAct = u.numR();
unsigned int numOpt = u.numC();
// w_j is row-vector of actor weights, for simple voting
assert(numAct == w.numC()); // require 1-to-1 matching of actors and strengths
assert(1 == w.numR()); // weights must be a row-vector
auto vfn = [vr, &w, &u](unsigned int k, unsigned int i, unsigned int j) {
double vkij = vote(vr, w(0, k), u(k, i), u(k, j));
return vkij;
};
auto c = coalitions(vfn, numAct, numOpt); // c(i,j) = strength of coaltion for i against j
KMatrix p = vProb(vpm, c); // p(i,j) = prob Ai defeats Aj
return p;
}
vector<KMatrix> VHCSearch::vn1(const KMatrix & m0, double s) {
unsigned int n = m0.numR();
auto nghbrs = vector<KMatrix>();
double pms[] = { -1, +1 };
for (unsigned int i = 0; i < n; i++) {
for (double si : pms) {
KMatrix m1 = m0;
m1(i, 0) = m0(i, 0) + (si*s);
nghbrs.push_back(m1);
}
}
return nghbrs;
}
tuple <KMatrix, VUI> RPState::pDist(int persp) const {
/// Calculate the probability distribution over states from this perspective
// TODO: convert this to a single, commonly used setup function
const unsigned int numA = model->numAct;
const unsigned int numP = numA; // for this demo, the number of positions is exactly the number of actors
// get unique indices and their probability
assert (0 < uIndices.size()); // should have been set with setUENdx();
//auto uNdx2 = uniqueNdx(); // get the indices to unique positions
const unsigned int numU = uIndices.size();
assert(numU <= numP); // might have dropped some duplicates
cout << "Number of aUtils: " << aUtil.size() << endl << flush;
const KMatrix u = aUtil[0]; // all have same beliefs in this demo
auto uufn = [u, this](unsigned int i, unsigned int j1) {
return u(i, uIndices[j1]);
};
auto uMat = KMatrix::map(uufn, numA, numU);
auto vpm = VPModel::Linear;
assert(uMat.numR() == numA); // must include all actors
assert(uMat.numC() == numU);
// vote_k ( i : j )
auto vkij = [this, uMat](unsigned int k, unsigned int i, unsigned int j) {
auto ak = (RPActor*)(model->actrs[k]);
auto v_kij = Model::vote(ak->vr, ak->sCap, uMat(k, i), uMat(k, j));
return v_kij;
};
// the following uses exactly the values in the given euMat,
// which may or may not be square
const KMatrix c = Model::coalitions(vkij, uMat.numR(), uMat.numC());
const KMatrix pv = Model::vProb(vpm, c); // square
const KMatrix p = Model::probCE(PCEModel::ConditionalPCM, pv); // column
const KMatrix eu = uMat*p; // column
assert(numA == eu.numR());
assert(1 == eu.numC());
return tuple <KMatrix, VUI>(p, uIndices);
}
// Given square matrix of Prob[i>j] returns a column vector for Prob[i].
// Uses 1-step conditional probabilities, not Markov process
KMatrix Model::condPCE(const KMatrix & pv) {
unsigned int numOpt = pv.numR();
auto p = KMatrix(numOpt, 1);
for (unsigned int i = 0; i < numOpt; i++) {
double pi = 1.0;
for (unsigned int j = 0; j < numOpt; j++) {
pi = pi * pv(i, j);
}
// double-check
assert(0 <= pi);
assert(pi <= 1);
p(i, 0) = pi; // probability that i beats all alternatives
}
double probOne = sum(p); // probability that one option, any option, beats all alternatives
p = (p / probOne); // conditional probability that i is that one.
return p;
}
// note that while the C_ij can be any arbitrary positive matrix
// with C_kk = 0, the p_ij matrix has the symmetry pij + pji = 1
// (and hence pkk = 1/2).
KMatrix Model::vProb(VPModel vpm, const KMatrix & c) {
unsigned int numOpt = c.numR();
assert(numOpt == c.numC());
auto p = KMatrix(numOpt, numOpt);
for (unsigned int i = 0; i < numOpt; i++) {
for (unsigned int j = 0; j < i; j++) {
double cij = c(i, j);
assert(0 <= cij);
double cji = c(j, i);
assert(0 <= cji);
assert((0 < cij) || (0 < cji));
auto ppr = vProb(vpm, cij, cji);
p(i, j) = get<0>(ppr); // set the lower left probability: if Linear, cij / (cij + cji)
p(j, i) = get<1>(ppr); // set the upper right probability: if Linear, cji / (cij + cji)
}
p(i, i) = 0.5; // set the diagonal probability
}
return p;
}
void RP2Model::setWeights(const KMatrix & w0) {
const unsigned int nr = w0.numR();
const unsigned int nc = w0.numC();
assert(1 == nr);
if (0 < numAct) {
assert(nc == numAct);
}
else {
numAct = nc;
}
for (auto w : w0) {
assert(0.0 <= w);
}
assert(minNumActor <= numAct);
assert(numAct <= maxNumActor);
// if it is OK, set it
wghtVect = w0;
return;
}
tuple<KMatrix, KMatrix> Model::probCE2(PCEModel pcm, VPModel vpm, const KMatrix & cltnStrngth) {
const double pTol = 1E-8;
unsigned int numOpt = cltnStrngth.numR();
auto p = KMatrix(numOpt, 1);
const auto victProb = Model::vProb(vpm, cltnStrngth); // prob of victory, square
switch (pcm) {
case PCEModel::ConditionalPCM:
p = condPCE(victProb);
break;
case PCEModel::MarkovIPCM:
p = markovIncentivePCE(cltnStrngth, vpm);
break;
case PCEModel::MarkovUPCM:
p = markovUniformPCE(victProb);
break;
default:
throw KException("Model::probCE unrecognized PCEModel");
break;
}
assert(numOpt == p.numR());
assert(1 == p.numC());
assert(fabs(sum(p) - 1.0) < pTol);
return tuple<KMatrix, KMatrix>(p, victProb);
}
RPState* RPState::doSUSN(ReportingLevel rl) const {
RPState* s2 = nullptr;
const unsigned int numA = model->numAct;
assert(numA == rpMod->actrs.size());
const unsigned int numU = uIndices.size();
assert ((0 < numU) && (numU <= numA));
assert (numA == eIndices.size());
// TODO: filter out essentially-duplicate positions
//printf("RPState::doSUSN: numA %i \n", numA);
//printf("RPState::doSUSN: numP %i \n", numP);
//cout << endl << flush;
const KMatrix u = aUtil[0]; // all have same beliefs in this demo
auto vpm = VPModel::Linear;
const unsigned int numP = pstns.size();
// Given the utility matrix, uMat, calculate the expected utility to each actor,
// as a column-vector. Again, this is from the perspective of whoever developed uMat.
auto euMat = [rl, numA, numP, vpm, this](const KMatrix & uMat) {
// BTW, be sure to lambda-bind uMat *after* it is modified.
assert(uMat.numR() == numA); // must include all actors
assert(uMat.numC() <= numP); // might have dropped some duplicates
auto uRng = [uMat](unsigned int i, unsigned int j) {
if ((uMat(i, j) < 0.0) || (1.0 < uMat(i, j))) {
printf("%f %i %i \n", uMat(i, j), i, j);
cout << flush;
cout << flush;
}
assert(0.0 <= uMat(i, j));
assert(uMat(i, j) <= 1.0);
return;
};
KMatrix::mapV(uRng, uMat.numR(), uMat.numC());
// vote_k ( i : j )
auto vkij = [this, uMat](unsigned int k, unsigned int i, unsigned int j) {
auto ak = (RPActor*)(rpMod->actrs[k]);
auto v_kij = Model::vote(ak->vr, ak->sCap, uMat(k, i), uMat(k, j));
return v_kij;
};
// the following uses exactly the values in the given euMat,
// which may or may not be square
const KMatrix c = Model::coalitions(vkij, uMat.numR(), uMat.numC());
const KMatrix pv = Model::vProb(vpm, c); // square
const KMatrix p = Model::probCE(PCEModel::ConditionalPCM, pv); // column
const KMatrix eu = uMat*p; // column
assert(numA == eu.numR());
assert(1 == eu.numC());
auto euRng = [eu](unsigned int i, unsigned int j) {
// due to round-off error, we must have a tolerance factor
const double tol = 1E-10;
const double euij = eu(i, j);
assert(0.0 <= euij+tol);
assert(euij <= 1.0+tol);
return;
};
KMatrix::mapV(euRng, eu.numR(), eu.numC());
if (ReportingLevel::Low < rl) {
printf("Util matrix is %i x %i \n", uMat.numR(), uMat.numC());
cout << "Assessing EU from util matrix: " << endl;
uMat.mPrintf(" %.6f ");
cout << endl << flush;
cout << "Coalition strength matrix" << endl;
c.mPrintf(" %12.6f ");
cout << endl << flush;
cout << "Probability Opt_i > Opt_j" << endl;
pv.mPrintf(" %.6f ");
cout << endl << flush;
cout << "Probability Opt_i" << endl;
p.mPrintf(" %.6f ");
cout << endl << flush;
cout << "Expected utility to actors: " << endl;
eu.mPrintf(" %.6f ");
cout << endl << flush;
}
return eu;
};
// end of euMat
auto euState = euMat(u);
cout << "Actor expected utilities: ";
KBase::trans(euState).mPrintf("%6.4f, ");
cout << endl << flush;
if (ReportingLevel::Low < rl) {
printf("--------------------------------------- \n");
printf("Assessing utility of actual state to all actors \n");
for (unsigned int h = 0; h < numA; h++) {
cout << "not available" << endl;
}
cout << endl << flush;
printf("Out of %u positions, %u were unique: ", numA, numU);
cout << flush;
for (auto i : uIndices) {
printf("%2i ", i);
}
cout << endl;
cout << flush;
}
auto uufn = [u, this](unsigned int i, unsigned int j1) {
return u(i, uIndices[j1]);
};
auto uUnique = KMatrix::map(uufn, numA, numU);
// Get expected-utility vector, one entry for each actor, in the current state.
const KMatrix eu0 = euMat(uUnique); // 'u' with duplicates, 'uUnique' without duplicates
s2 = new RPState(model);
//s2->pstns = vector<KBase::Position*>();
for (unsigned int h = 0; h < numA; h++) {
s2->pstns.push_back(nullptr);
}
// TODO: clean up the nesting of lambda-functions.
// need to create a hypothetical state and run setOneAUtil(h,Silent) on it
//
// The newPosFn does a GA optimization to find the best next position for actor h,
// and stores it in s2. To do that, it defines three functions for evaluation, neighbors, and show:
// efn, nfn, and sfn.
auto newPosFn = [this, rl, euMat, u, eu0, s2](const unsigned int h) {
s2->pstns[h] = nullptr;
auto ph = ((const MtchPstn *)(pstns[h]));
// Evaluate h's estimate of the expected utility, to h, of
// advocating position mp. To do this, build a hypothetical utility matrix representing
// h's estimates of the direct utilities to all other actors of h adopting this
// Position. Do that by modifying the h-column of h's matrix.
// Then compute the expected probability distribution, over h's hypothetical position
// and everyone else's actual position. Finally, compute the expected utility to
// each actor, given that distribution, and pick out the value for h's expected utility.
// That is the expected value to h of adopting the position.
auto efn = [this, euMat, rl, u, h](const MtchPstn & mph) {
// This correctly handles duplicated/unique options
// We modify the given euMat so that the h-column
// corresponds to the given mph, but we need to prune duplicates as well.
// This entails some type-juggling.
const KMatrix uh0 = aUtil[h];
assert(KBase::maxAbs(u - uh0) < 1E-10); // all have same beliefs in this demo
if (mph.match.size() != rpMod->numItm) {
cout << mph.match.size() << endl << flush;
cout << rpMod->numItm << endl << flush;
cout << flush << flush;
}
assert(mph.match.size() == rpMod->numItm);
auto uh = uh0;
for (unsigned int i = 0; i < rpMod->numAct; i++) {
auto ai = (RPActor*)(rpMod->actrs[i]);
double uih = ai->posUtil(&mph);
uh(i, h) = uih; // utility to actor i of this hypothetical position by h
}
// 'uh' now has the correct h-column. Now we need to see how many options
// are unique in the hypothetical state, and keep only those columns.
// This entails juggling back and forth between the all current positions
// and the one hypothetical position (mph at h).
// Thus, the next call to euMat will consider only unique options.
auto equivHNdx = [this, h, mph](const unsigned int i, const unsigned int j) {
// this little function takes care of the different types needed to compare
// dynamic pointers to positions (all but h) with a constant position (h itself).
// In other words, the comparisons for index 'h' use the hypothetical mph, not pstns[h]
bool rslt = false;
auto mpi = ((const MtchPstn *)(pstns[i]));
auto mpj = ((const MtchPstn *)(pstns[j]));
assert(mpi != nullptr);
assert(mpj != nullptr);
if (i == j) {
rslt = true; // Pi == Pj, always
}
else if (h == i) {
rslt = (mph == (*mpj));
}
else if (h == j) {
rslt = ((*mpi) == mph);
}
else {
rslt = ((*mpi) == (*mpj));
}
return rslt;
};
auto ns = KBase::uiSeq(0, model->numAct - 1);
const VUI uNdx = get<0>(KBase::ueIndices<unsigned int>(ns, equivHNdx));
const unsigned int numU = uNdx.size();
auto hypUtil = KMatrix(rpMod->numAct, numU);
// we need now to go through 'uh', copying column J the first time
// the J-th position is determined to be equivalent to something in the unique list
for (unsigned int i = 0; i < rpMod->numAct; i++) {
for (unsigned int j1 = 0; j1 < numU; j1++) {
unsigned int j2 = uNdx[j1];
hypUtil(i, j1) = uh(i, j2); // hypothetical utility in column h
}
}
if (false) {
cout << "constructed hypUtil matrix:" << endl << flush;
hypUtil.mPrintf(" %8.2f ");
cout << endl << flush;
}
if (ReportingLevel::Low < rl) {
printf("--------------------------------------- \n");
printf("Assessing utility to %2i of hypo-pos: ", h);
printPerm(mph.match);
cout << endl << flush;
printf("Hypo-util minus base util: \n");
(uh - uh0).mPrintf(" %+.4E ");
cout << endl << flush;
}
const KMatrix eu = euMat(hypUtil); // uh or hypUtil
// BUG: If we use 'uh' here, it passes the (0 <= delta-EU) test, because
// both hypothetical and actual are then calculated without dropping duplicates.
// If we use 'hypUtil' here, it sometimes gets (delta-EU < 0), because
// the hypothetical drops duplicates but the actual (computed elsewhere) does not.
// FIX: fix the 'elsewhere'
const double euh = eu(h, 0);
assert(0 < euh);
//cout << euh << endl << flush;
//printPerm(mp.match);
//cout << endl << flush;
//cout << flush;
return euh;
}; // end of efn
/*
// I do not actually use prevMP, but it is still an example for std::set
auto prevMP = [](const MtchPstn & mp1, const MtchPstn & mp2) {
bool r = std::lexicographical_compare(
mp1.match.begin(), mp1.match.end(),
mp2.match.begin(), mp2.match.end());
return r;
};
std::set<MtchPstn, bool(*)(const MtchPstn &, const MtchPstn &)> mpSet(prevMP);
*/
// return vector of neighboring 1-permutations
auto nfn = [](const MtchPstn & mp0) {
const unsigned int numI = mp0.match.size();
auto mpVec = vector <MtchPstn>();
mpVec.push_back(MtchPstn(mp0));
// one-permutations
for (unsigned int i = 0; i < numI; i++) {
for (unsigned int j = i + 1; j < numI; j++) {
unsigned int ei = mp0.match[i];
unsigned int ej = mp0.match[j];
auto mij = MtchPstn(mp0);
mij.match[i] = ej;
mij.match[j] = ei;
mpVec.push_back(mij);
}
}
// two-permutations
for (unsigned int i = 0; i < numI; i++) {
for (unsigned int j = i + 1; j < numI; j++) {
for (unsigned int k = j + 1; k < numI; k++) {
unsigned int ei = mp0.match[i];
unsigned int ej = mp0.match[j];
unsigned int ek = mp0.match[k];
auto mjki = MtchPstn(mp0);
mjki.match[i] = ej;
mjki.match[j] = ek;
mjki.match[k] = ei;
mpVec.push_back(mjki);
auto mkij = MtchPstn(mp0);
mkij.match[i] = ek;
mkij.match[j] = ei;
mkij.match[k] = ej;
mpVec.push_back(mkij);
}
}
}
//unsigned int mvs = mpVec.size() ;
//cout << mvs << endl << flush;
//cout << flush;
return mpVec;
}; // end of nfn
// show some representation of this position on cout
auto sfn = [](const MtchPstn & mp0) {
printPerm(mp0.match);
return;
};
auto ghc = new KBase::GHCSearch<MtchPstn>();
ghc->eval = efn;
ghc->nghbrs = nfn;
ghc->show = sfn;
auto rslt = ghc->run(*ph, // start from h's current positions
ReportingLevel::Silent,
100, // iter max
3, 0.001); // stable-max, stable-tol
if (ReportingLevel::Low < rl) {
printf("---------------------------------------- \n");
printf("Search for best next-position of actor %2i \n", h);
//printf("Search for best next-position of actor %2i starting from ", h);
//trans(*aPos).printf(" %+.6f ");
cout << flush;
}
double vBest = get<0>(rslt);
MtchPstn pBest = get<1>(rslt);
unsigned int iterN = get<2>(rslt);
unsigned int stblN = get<3>(rslt);
delete ghc;
ghc = nullptr;
if (ReportingLevel::Medium < rl) {
printf("Iter: %u Stable: %u \n", iterN, stblN);
printf("Best value for %2i: %+.6f \n", h, vBest);
cout << "Best position: " << endl;
cout << "numCat: " << pBest.numCat << endl;
cout << "numItm: " << pBest.numItm << endl;
cout << "perm: ";
printPerm(pBest.match);
cout << endl << flush;
}
MtchPstn * posBest = new MtchPstn(pBest);
s2->pstns[h] = posBest;
// no need for mutex, as s2->pstns is the only shared var,
// and each h is different.
double du = vBest - eu0(h, 0); // (hypothetical, future) - (actual, current)
if (ReportingLevel::Low < rl) {
printf("EU improvement for %2i of %+.4E \n", h, du);
}
//printf(" vBest = %+.6f \n", vBest);
//printf(" eu0(%i, 0) for %i = %+.6f \n", h, h, eu0(h,0));
//cout << endl << flush;
// Logically, du should always be non-negative, as GHC never returns a worse value than the starting point.
// However, actors plan on the assumption that all others do not change - yet they do.
const double eps = 0.05; // 0.025; // enough to avoid problems with round-off error
assert(-eps <= du);
return;
}; // end of newPosFn
const bool par = true;
auto ts = vector<thread>();
// Each actor, h, finds the position which maximizes their EU in this situation.
for (unsigned int h = 0; h < numA; h++) {
if (par) { // launch all, concurrent
ts.push_back(thread([newPosFn, h]() {
newPosFn(h);
return;
}));
}
else { // do each, sequential
newPosFn(h);
}
}
if (par) { // now join them all before continuing
for (auto& t : ts) {
t.join();
}
}
assert(nullptr != s2);
assert(numP == s2->pstns.size());
assert(numA == s2->model->numAct);
for (auto p : s2->pstns) {
assert(nullptr != p);
}
s2->setUENdx();
return s2;
}
// return the list of the most self-interested position of each actor,
// with the CP last.
// As a side-affect, set each actor's min/max permutation values so as to
// compute normalized utilities later.
vector<VUI> scanAllPossiblePositions(const RPModel * rpm) {
unsigned int numA = rpm->numAct;
unsigned int numRefItem = rpm->numItm;
assert(numRefItem == rpm->numCat);
LOG(INFO) << "There are" << numA << "actors and" << numRefItem << "reform items";
KMatrix aCap = KMatrix(1, numA);
for (unsigned int i = 0; i < numA; i++) {
auto ri = ((const RPActor *)(rpm->actrs[i]));
aCap(0, i) = ri->sCap;
}
LOG(INFO) << "Actor capabilities: ";
aCap.mPrintf(" %.2f ");
LOG(INFO) << "Effective gov cost of items:";
(rpm->govCost).mPrintf("%.3f ");
LOG(INFO) << "Government budget: " << rpm->govBudget;
assert(0 < rpm->govBudget);
string log("Value to actors (rows) of individual reform items (columns):");
for (unsigned int i = 0; i < rpm->actrs.size(); i++) {
auto rai = ((const RPActor*)(rpm->actrs[i]));
for (unsigned int j = 0; j < numRefItem; j++) {
double vij = rai->riVals[j];
log += KBase::getFormattedString(" %6.2f", vij);
}
}
LOG(INFO) << log;
LOG(INFO) << "Computing positions ... ";
vector<VUI> allPositions; // list of all possiblepositions
VUI pstn;
// build the first permutation: 0,1,2,3,...
for (unsigned int i = 0; i < numRefItem; i++) {
pstn.push_back(i);
}
allPositions.push_back(pstn);
while (next_permutation(pstn.begin(), pstn.end())) {
allPositions.push_back(pstn);
}
const unsigned int numPos = allPositions.size();
LOG(INFO) << "For" << numRefItem << "reform items there are"
<< numPos << "positions";
// -------------------------------------------------
// The next section sets up actor utilities.
// First, we compute the unnormalized, raw utilities. The 'utilActorPos' checks
// to see if pvMin/pvMax have been set, and returns the raw scores if not.
// Then we scan across rows to find that actor's pvMin/pvMax, and record that
// so utilActorPos can use it in the future. Finally, we normalize the rows and
// display the normalized utility matrix.
auto ruFn = [allPositions, rpm](unsigned int ai, unsigned int pj) {
auto pstn = allPositions[pj];
double uip = rpm->utilActorPos(ai, pstn);
return uip;
};
LOG(INFO) << "Computing utilities of positions ... ";
// rows are actors, columns are all possible positions
auto rawUij = KMatrix::map(ruFn, numA, numPos);
// set the min/max for each actor
for (unsigned int i = 0; i < numA; i++) {
double pvMin = rawUij(i, 0);
double pvMax = rawUij(i, 0);
for (unsigned int j = 0; j < numPos; j++) {
double rij = rawUij(i, j);
if (rij < pvMin) {
pvMin = rij;
}
if (rij > pvMax) {
pvMax = rij;
}
}
assert(0 <= pvMin);
assert(pvMin < pvMax);
auto ai = ((RPActor*)(rpm->actrs[i]));
ai->posValMin = pvMin;
ai->posValMax = pvMax;
}
LOG(INFO) << "Normalizing utilities of positions ... ";
KMatrix uij = KBase::rescaleRows(rawUij, 0.0, 1.0); // von Neumann utility scale
string utilMtx("Complete (normalized) utility matrix of all possible positions (rows) versus actors (columns) \n");
for (unsigned int pj = 0; pj < numPos; pj++) {
utilMtx += KBase::getFormattedString("%3u ", pj);
auto pstn = allPositions[pj];
//printVUI(pstn);
utilMtx += KBase::stringVUI(pstn);
utilMtx += " ";
for (unsigned int ai = 0; ai < numA; ai++) {
double uap = uij(ai, pj);
utilMtx += KBase::getFormattedString("%6.4f, ", uap);
}
utilMtx += KBase::getFormattedString("\n");
}
LOG(INFO) << utilMtx;
// -------------------------------------------------
// The next section determines the most self-interested positions for each actor,
// as well as the 'central position' over all possible reform priorities
// (which 'office seeking politicans' would adopt IF proportional voting).
LOG(INFO) << "Computing best position for each actor";
vector<VUI> bestAP; // list of each actor's best position (followed by CP)
for (unsigned int ai = 0; ai < numA; ai++) {
unsigned int bestJ = 0;
double bestV = 0;
for (unsigned int pj = 0; pj < numPos; pj++) {
if (bestV < uij(ai, pj)) {
bestJ = pj;
bestV = uij(ai, pj);
}
}
string bestMtx("Best position for ");
string ais = std::to_string(ai);
//string bjs = std::to_string(bestJ);
string ps = KBase::stringVUI(allPositions[bestJ]);
bestMtx += ais + " is " + ps;
LOG(INFO) << bestMtx;
//LOG(INFO) << "Best for" << ai << "is ";
//printVUI(positions[bestJ]);
bestAP.push_back(allPositions[bestJ]);
}
LOG(INFO) << "Computing zeta ... ";
KMatrix zeta = aCap * uij;
assert((1 == zeta.numR()) && (numPos == zeta.numC()));
LOG(INFO) << "Sorting positions from most to least net support ...";
auto betterPR = [](tuple<unsigned int, double, VUI> pr1,
tuple<unsigned int, double, VUI> pr2) {
double v1 = get<1>(pr1);
double v2 = get<1>(pr2);
bool better = (v1 > v2);
return better;
};
auto pairs = vector<tuple<unsigned int, double, VUI>>();
for (unsigned int i = 0; i < numPos; i++) {
auto pri = tuple<unsigned int, double, VUI>(i, zeta(0, i), allPositions[i]);
pairs.push_back(pri);
}
sort(pairs.begin(), pairs.end(), betterPR);
const unsigned int maxDisplayed = 720; // factorial(6)
unsigned int numPr = (pairs.size() < maxDisplayed) ? pairs.size() : maxDisplayed;
LOG(INFO) << "Displaying highest" << numPr;
for (unsigned int i = 0; i < numPr; i++) {
auto pri = pairs[i];
unsigned int ni = get<0>(pri);
double zi = get<1>(pri);
VUI pi = get<2>(pri);
string ps = KBase::stringVUI(pi);
LOG(INFO) << KBase::getFormattedString(" %3u: %4u %.2f %s", i, ni, zi, ps.c_str());
//printVUI(pi);
}
VUI bestPerm = get<2>(pairs[0]);
bestAP.push_back(bestPerm);
return bestAP;
}
// Given square matrix of strengths, Coalition[i over j] returns a column vector for Prob[i].
// Uses Markov process, not 1-step conditional probability.
// Challenge probabilities are proportional to influence promoting a challenge
KMatrix Model::markovIncentivePCE(const KMatrix & coalitions, VPModel vpm) {
using KBase::sqr;
using KBase::qrtc;
const bool printP = false;
const double pTol = 1E-8;
const unsigned int numOpt = coalitions.numR();
assert(numOpt == coalitions.numC());
const auto victProbMatrix = vProb(vpm, coalitions);
// given coalitions, calculate the total incentive for i to challenge j
// This is n[ i -> j] in the "Markov Voting with Incentives in KTAB" paper
auto iFn = [victProbMatrix, coalitions](unsigned int i, unsigned int j) {
const double epsSupport = 1E-10;
const double sij = coalitions(i, j);
double inctv = sij * victProbMatrix(i,j);
if (i == j) {
inctv = inctv + epsSupport;
}
return inctv;
};
const auto inctvMatrix = KMatrix::map(iFn, numOpt, numOpt);
// Using the incentives, calculate the probability of i challenging j,
// given that j is the current favorite proposal.
// This is P[ i -> j] in the "Markov Voting with Incentives in KTAB" paper
// Note that if every actor prefers j to every other option,
// then all incentive(i,j) will be zero, except incentive(j,j) = eps.
// Even in this case, we will not get a division by zero error,
// and it will correctly return that the only "challenger" is j itself,
// with guaranteed success.
//
auto cpFn = [inctvMatrix, numOpt](unsigned int i, unsigned int j) {
double sum = 0.0;
for (unsigned int k = 0; k < numOpt; k++) {
sum = sum + inctvMatrix(k, j);
}
const double pij = inctvMatrix(i, j) / sum;
return pij;
};
const auto chlgProbMatrix = KMatrix::map(cpFn, numOpt, numOpt);
// probability starts as uniform distribution (column vector)
auto p = KMatrix(numOpt, 1, 1.0) / numOpt; // all 1/n
auto q = p;
unsigned int iMax = 1000; // 10-30 is typical
unsigned int iter = 0;
double change = 1.0;
// do the markov calculation
while (pTol < change) { // && (iter < iMax)
if (printP) {
printf("Iteration %u / %u \n", iter, iMax);
cout << "pDist:" << endl;
trans(p).mPrintf(" %.4f");
cout << endl;
printf("change: %.4e \n", change);
cout << endl << flush;
}
auto ct = KMatrix(numOpt, numOpt);
for (unsigned int i = 0; i < numOpt; i++) {
for (unsigned int j = 0; j < numOpt; j++) {
// See "Markov Voting with Incentives in KTAB" paper
ct(i, j) = p(i, 0) * chlgProbMatrix(j, i);
}
}
if (printP) {
cout << "Ct:" << endl;
ct.mPrintf(" %.3f");
cout << endl << flush;
}
change = 0.0;
for (unsigned int i = 0; i < numOpt; i++) {
double qi = 0.0;
for (unsigned int j = 0; j < numOpt; j++) {
double vij = victProbMatrix(i, j);
double cj = ct(i, j) + ct(j, i);
qi = qi + vij* cj;
}
assert(0 <= qi); // double-check
q(i, 0) = qi;
double c = fabs(q(i, 0) - p(i, 0));
change = (c > change) ? c : change;
}
// Newton method improves convergence.
p = (p+q)/2.0;
iter++;
assert(fabs(sum(p) - 1.0) < pTol); // double-check
}
assert(iter < iMax); // no way to recover
return p;
}