DESARROLLO DE CIUDADANÍA PARTICIPAN Y EMPODERAMIENTO EN SALUD Proceso por el cual los grupos sociales e individuales asumen y expresan

During the course of this thesis, we have spoken on various occasions about dynamic policies as opposed to pre-computed or static policies. Similarly, we have spoken about stochastic and deterministic policies. What does this mean, what is the difference and what are the implications?

Consider the well-known rock-paper-scissors game. In this game, non-random be-havior can be exploited, and thus the safest way to protect oneself against an opponent consists in playing with uniform probability¹. Suppose we have to play a sequence of rock-paper-scissors games. The uniform strategy can be implemented statically, e.g.

by sampling each move beforehand and revealing them turn by turn; or, it can be implemented dynamically, e.g. by deciding each move just before it has to be played². This flexibility does not exists if we wanted to use a deterministic strategy, because by choosing a deterministic policy we are determining all the moves beforehand. In general, having uncertainty allows us determining the value of random variables dynamically.

9.1.1 Risk versus Ambiguity

A special case occurs when there is uncertainty over the very beliefs of the decision maker, as is the case modeled by the Bayesian I/O model introduced in 3.2.2. The author proposes that this kind of uncertainty relates to an old debate dating from the early conceptualizations of decision theory: riskversus ambiguity(Knight, 1921).

Risk corresponds to the odds that the decision maker has adequate knowledge of, whereas ambiguity to the odds that are unknown to him. From the two, risk is well-understood; indeed, classical decision theory was regarded by its very founders as a

“normative foundation of optimal decision making under risk” (von Neumann and Morgenstern, 1944; Savage, 1954). The success of these frameworks had put under question the operational relevance of the concept of ambiguity, until Ellsberg presented the paradox named after him in his seminal paper (Ellsberg, 1961). Several mathe-matical formalizations for ambiguity have been proposed in the literature, but none of them has found widespread acceptance (Camerer and Weber, 1992). However, on an abstract level the vast majority of approaches considers ambiguity to be some kind of uncertainty about risk or uncertainty of belief or lack of information. Here we propose a simple formalization that is an original contribution of this thesis.

To illustrate the formalization, let us review an example. Imagine a rational decision maker has to place a bet over the outcome of a biased coin toss which is either Head

1In game-theoretic parlance, it is said that the rock-paper-scissors game has a mixed strategyNash equilibrium(Osborne and Rubinstein, 1999).

2In computer science, this technique of delaying the evaluation of a quantity up until the point where it is needed is known aslazy evaluation (Pratt and Zelkowitz, 2000).

or Tail. The payoff is $1 for a correct bet or $0 for a wrong bet. Given the rational decision maker’s belief, we want to predict the bet he will place under five different cases, illustrated in Figure 9.1:

I. He believes that the odds are ¹₄ for Head and ³₄ for Tail.

II. He believes that the odds are ³₄ for Head and ¹₄ for Tail.

III. He believes that the odds ⁵₈ for Head and ³₈ for Tail.

IV. He believes that either I or II occurs with probability ¹₄ or ³₄ respectively.

V. Hefinds himself believing in either I or II with probability ¹₄ or ³₄ respectively.

Cases I–IV are easily examined under the framework of maximum expected utility. The rational decision maker places the bet that maximizes his expected payoff. For cases I and II, the optimal bets are Tail and Head respectively, because their expected payoff is

$0.75, as opposed to $0.25 offered by the alternative bet (Figures 9.1a & b). Likewise, in III the decision maker bets Head. In case IV, the decision maker can reexpress the two-stage coin toss, i.e. first selecting between situation I or II and then tossing the coin (Figure 9.1c), as an equivalent single-stage coin toss with a re-weighted bias obtained by multiplying the probabilities of the first stage with the probabilities of the second stage (Figure 9.1e). This reduction reveals that case IV is equivalent to case III, with Head being to optimal bet.

However, by construction, case V requires a different analysis. Here, the decision maker’s belief can take on one out of two possible forms, in which the optimal bets are Tail and Head as discussed previously. The crucial difference lies in the fact that the probabilities of the belief instantiations are beyond the scope of the decision maker’s analysis. Therefore, for him it is optimal to bet Tail when reaching case I and to bet Head when reaching case II. This innocuous fact is by no means trivial, because the subjective expected utility of the decision maker is ¹₄·³₄+³₄·³₄ = ³₄ > ⁵₈, that is,strictly higher than the subjective expected utility of the classical analysis—and in practice, subjective beliefs are all what a decision maker has³! We call these probabilities that are exogenous to the decision maker’s beliefsambiguities.

The distinction has an operational meaning. Notice that in case V, the belief of the decision maker is itself a random variable, implying that the optimal policy is undefined until the random variable is resolved. Hence, the computation of the optimal policy can be delayed, i.e. the optimal policy can be determined dynamically. This is unlike case IV, where the policy is pre-computed/static. The corresponding Bayesian I/O model is as follows. Letθ∈ {I, II}be the parameter determining whether the decision maker

3Intuitively, it seems safer to delay one’s decision until the evidence is conclusive.

I II III

IV V

H H H

H H

H H

T T T

T T

T T

1 4 1

4 1

4

3 4 3

4 3

4

3 4 3

4

3 4

1 4 1

4

1 4

5 8

3 8

1 4 1

4

3 4 3

4

T T

H

H H

H

Figure 9.1: Risk versus Ambiguity. In the figure, five different decision making scenarios are shown. A biased coin is tossed. The goal is to predict the outcome and the payoffs are $1 and

$0 for a right and wrong guess respectively. A rational decision maker places bets (here shown inside speech bubbles) such that his subjective expected utility is maximized. These subjective beliefs are delimited within dotted boxes. The cases in panels I–IV differ from case V in that the former can be fully understood in terms of classical decision theory, whereas the latter cannot.

is in case I or II. Then, the full Bayesian I/O model is given by:

P(θ= I) = 1

4 P(θ= II) = 3

4 P(a|θ= I) =

(0 if a= H

1 if a= T P(a|θ= II) =

(1 ifa= H 0 ifa= T P(o|θ= I, a) =

(1

4 ifo= H

3

4 ifo= T P(o|θ= II, a) = (3

4 if o= H

1

4 if o= T.

Here, the prior probabilities P(θ) are ambiguities while the P(a|θ) and P(o|θ, a) are risk probabilities, because fixing θ determines the decision maker’s estimation about the outcome and his policy.