The Principal-Agent Problem for Delegation-as-Prediction

This model is a follow-up to Scoring Rules, extending the analysis of truthful prediction to settings where an agent’s report can influence, rather than merely predict, which event actually occurs.

The agent gets reward based on which event happens. As long as the agent’s prediction doesn’t influence what event happens (full supervision), it will continue to make truthful predictions:

///fold:
var Infer = function(thunk){
  return Enumerate(thunk);
}

var Expectation = function(thunk){
  return expectation(Infer(thunk));
}

var makeCategorical = function(vs, ps){
  return window.makeCategoricalERP(ps, vs);
};

var prob = function(erp, val) {
  return Math.exp(erp.score([], val))
}

var squared = function(x) {
  return Math.pow(x, 2)
}

var distPrior = Enumerate(function(){
  var potentialPs = [
    [.5, .3, .2],
    [.3, .5, .2],
    [.2, .3, .5],
    [0.005, .99, 0.005]
  ];
  var ps = uniformDraw(potentialPs);
  return makeCategorical(['A', 'B', 'C'], ps);
});

var alpha = 100;

var logScore = function(dist, event) {
  return Math.log(prob(dist, event));
}
///

var getReportedDistForParams = function(params) {

  var predictionScore = logScore;  
  
  var eventScore = function(event) {
    return event == 'A' ? params.eventRewardA : 0;
  }

  var agent = function(){
    return Infer(function(){
      var beliefDist = makeCategorical(['A', 'B', 'C'], [.3, .5, .2]);
      var reportDist = sample(distPrior);
      var expectedScore = Expectation(function(){
        var event = sample(beliefDist);
        return predictionScore(reportDist, event) + eventScore(event);
      });
      factor(alpha * expectedScore);
      return reportDist;
    });
  }

  return agent().MAP().val;

};

print('Reported distribution with full supervision, event-utility for A:');
print(getReportedDistForParams({
  eventRewardA: 5
}))

We move to partial (stochastic) supervision, and in the unsupervised case, it is the agent’s prediction that determines what happens. Now utilities skew predictions:

///fold:
var Infer = function(thunk){
  return Enumerate(thunk);
}

var Expectation = function(thunk){
  return expectation(Infer(thunk));
}

var makeCategorical = function(vs, ps){
  return window.makeCategoricalERP(ps, vs);
};

var prob = function(erp, val) {
  return Math.exp(erp.score([], val))
}

var squared = function(x) {
  return Math.pow(x, 2)
}

var distPrior = Enumerate(function(){
  var potentialPs = [
    [.5, .3, .2],
    [.3, .5, .2],
    [.2, .3, .5],
    [0.005, .99, 0.005]
  ];
  var ps = uniformDraw(potentialPs);
  return makeCategorical(['A', 'B', 'C'], ps);
});

var alpha = 100;

var logScore = function(dist, event) {
  return Math.log(prob(dist, event));
}
///

var getReportedDistForParams = function(params) {

  var predictionScore = logScore;  
  
  var eventScore = function(event) {
    return event == 'A' ? params.eventRewardA : 0;
  }

  var agent = function(){
    return Infer(function(){      
      var beliefDist = makeCategorical(['A', 'B', 'C'], [.3, .5, .2]);
      var reportDist = sample(distPrior);
      var expectedScore = Expectation(function(){
        var isSupervised = flip(params.probOfSupervision);
        if (isSupervised) {
          var event = sample(beliefDist);
          return predictionScore(reportDist, event) + eventScore(event);
        } else {
          var event = sample(reportDist); // enact what agent predicts will happen
          return eventScore(event);
        }
      });
      factor(alpha * expectedScore);
      return reportDist;
    });
  }

  return agent().MAP().val;

};

var exploreRewardParams = function(){  
  var rewards = [0, 0.0001, 0.001, 0.01, 0.1, 1, 10]
  map(
    function(reward){
      var reportDist = getReportedDistForParams({
        eventRewardA: reward,
        probOfSupervision: .1
      });
      var cheated = reportDist.score([], 'A') != Math.log(.3);
      print([reward, cheated ? 'cheated' : 'honest']);
    },
    rewards);
};

var exploreSupervisionParams = function(){  
  var supervisionProbs = [0.01, 0.1, 0.3, 0.5, 1]
  map(
    function(probOfSupervision){
      var reportDist = getReportedDistForParams({
        eventRewardA: .1,
        probOfSupervision: probOfSupervision
      });
      var cheated = reportDist.score([], 'A') != Math.log(.3);
      print([probOfSupervision, cheated ? 'cheated' : 'honest']);
    },
    supervisionProbs);
};


print('Reported distribution with partial supervision, event-utility for A:');
print(getReportedDistForParams({
  eventRewardA: 5,
  probOfSupervision: .1
}))

print('\nAs reward for action "A" increases, probability of cheating increases:');
exploreRewardParams()

print('\nAs probability of supervision increases, probability of cheating decreases:');
exploreSupervisionParams()