The impact of statistical learning on violations of the sure-thing principle

Abstract

This paper experimentally tests whether violations of Savage’s (1954) sure-thing principle (STP) decrease through statistical learning. Our subjects repeatedly had to bet on the drawings from an urn with an unknown proportion of differently colored balls. The control group was thereby subjected to learning through mere thought only. In addition, the test group received more and more statistical information over the course of the experiment by observing the color of the ball actually drawn after each bet. We expected that statistical learning would decrease the decision makers’ ambiguity, thereby implying a stronger decrease of STP violations in the test than in the control group. However, our data surprisingly shows that learning by mere thought rather than statistical learning leads to a decrease in STP violations.

Appendices

Appendix A: The sure-thing principle

The Savage framework considers a state space S, a set of outcomes X, and a set of acts F which map the state space into the outcome space, i.e. f:S→X for f∈F. Savage (1954) provides a set of axioms over preferences $\succsim$ on the acts in F which gives rise to his famous subjective EU theory. His key axiom for deriving subjective EU theory is the sure-thing principle (STP) which formally states that for all acts f,g,h,h ^′∈F and all events E⊂S,

$$ f_{E}h\succsim g_{E}h\text{ implies}\ \ f_{E}h^{\prime }\succsim g_{E}h^{\prime }\text{.} $$

(5)

Here f _E h denotes the act that gives the outcomes of act f in event E and the outcomes of act h else, i.e.

$$ f_{E}h\left( s\right) =\left\{ \begin{array}{cc} f\left( s\right) & \text{for }s\in E \\ h\left( s\right) & \text{for }s\in S\backslash E\text{.} \end{array} \right. $$

(6)

To formally describe the choice between the prospects A and B, resp. A ^′ and B ^′, within the Savage framework, let us reinterpret these prospects as the following acts

$$\begin{array}{@{}rcl@{}} \mathbf{A} &=&f_{Y\cup B}h\text{, }\mathbf{B}=g_{Y\cup B}h\text{;} \end{array} $$

(7)

$$\begin{array}{@{}rcl@{}} \mathbf{A}^{\prime } &=&f_{Y\cup B}h^{\prime }\text{, }\mathbf{B}^{\prime }=g_{Y\cup B}h^{\prime }\end{array} $$

(8)

such that

$$ f\left( s\right) =\left\{ \begin{array}{cc} x & \text{for }s\in Y \\ z & \text{for }s\in B \end{array} \right. $$

(9)

as well as g(s)=y for s∈Y∪B and h(s)=y , h ^′(s)=x for s∈R. For example, the revealed choice pair A,B ^′, i.e.

$$ \mathbf{A\succ B}\text{ and }\mathbf{B}^{\prime }\mathbf{\succ A}^{\prime } \text{,} $$

(10)

then corresponds to the following revealed preferences over Savage acts

$$ f_{Y\cup B}h\mathbf{\succ }g_{Y\cup B}h\text{ and }g_{Y\cup B}h^{\prime } \mathbf{\succ }f_{Y\cup B}h^{\prime }\text{,} $$

(11)

which violate the STP. Similarly, it is straightforward to see that B,A ^′ also violates the STP whereas the choice pairs A,A ^′ and B,B ^′ are consistent with the STP.

Appendix B: Prospect theory

Although the STP violating choice pairs (2) cannot be represented by EU theory, they can be represented by prospect theory. Restricted to the domain of gains, prospect theory gives rise to the following utility representation for preferences $\succsim$ over Savage acts f,g∈F,

$$ f\succsim g\Leftrightarrow \int\limits_{s\in S}^{C}u\left( f\left( s\right) \right) d\nu \left( s\right) \geq \int\limits_{s\in S}^{C}u\left( g\left( s\right) \right) d\nu \left( s\right) $$

(12)

where ν is a unique non-additive (=not necessarily additive) probability measure satisfying ν(∅)=0, ν(S)=1, and ν(E)≤ν(E ^′) if E⊂E ^′; and the integral in (12) is the Choquet integral.⁸

$$ \int\limits_{s\in S}^{C}u\left( f\left( s\right) \right) d\nu \left( s\right) = \sum\limits_{i=1}^{m}u\left( f\left( s_{i}\right) \right) \cdot \left[ \nu \left( E_{1}\cup ...\cup E_{i}\right) -\nu \left( E_{1}\cup ...\cup E_{i-1}\right) \right] $$

(13)

Focus at first on the choice pair A,B ^′. Without loss of generality, set u(z)=1 and u(x)=0 so that we obtain for a prospect theory decision maker that

$$\begin{array}{@{}rcl@{}} \mathbf{A} &\succ &\mathbf{B}\Leftrightarrow \end{array} $$

(14)

$$\begin{array}{@{}rcl@{}} \int\limits_{s\in S}^{C}u\left( f_{Y\cup B}h\right) dv\left( s\right) &>&\int\limits_{s\in S}^{C}u\left( g_{Y\cup B}h\right) dv\left( s\right) \Leftrightarrow \end{array} $$

(15)

$$\begin{array}{@{}rcl@{}} \nu \left( B\right) +u\left( y\right) \left[ \nu \left( B\cup R\right) -\nu \left( B\right) \right] &>&u\left( y\right)\end{array} $$

(16)

as well as

$$\begin{array}{@{}rcl@{}} \mathbf{B}^{\prime } &\succ &\mathbf{A}^{\prime }\Leftrightarrow \end{array} $$

(17)

$$\begin{array}{@{}rcl@{}} \int\limits_{s\in S}^{C}u\left( g_{Y\cup B}h^{\prime }\right) dv\left( s\right) &>&\int\limits_{s\in S}^{C}u\left( f_{Y\cup B}h^{\prime }\right) dv\left( s\right) \Leftrightarrow \end{array} $$

(18)

$$\begin{array}{@{}rcl@{}} u\left( y\right) \left[ \nu \left( B\cup Y\right) \right] &>&\nu \left( B\right) \text{.} \end{array} $$

(19)

Combining inequalities (16) and (19) shows that the utility representation (12) of the revealed choices A,B ^′ requires local concavity of ν in the following sense⁹

$$ \nu \left( E\cup E^{\prime }\right) +\nu \left( E\cap E^{\prime }\right) \leq \left( \text{resp. }\geq \right) \nu \left( E\right) +\nu \left( E^{\prime }\right) \text{.} $$

(20)

$$\begin{array}{@{}rcl@{}} u\left( y\right) \left[ \nu \left( B\cup Y\right) \right] &>&u\left( y\right) \left[ 1-\left[ \nu \left( B\cup R\right) -\nu \left( B\right) \right] \right] \Leftrightarrow \end{array} $$

(21)

$$\begin{array}{@{}rcl@{}} \nu \left( B\cup Y\right) &>&\nu \left( S\right) -\left[ \nu \left( B\cup R\right) -\nu \left( B\right) \right] \Leftrightarrow \end{array} $$

(22)

$$\begin{array}{@{}rcl@{}} \nu \left( B\cup Y\right) \!+\nu \left( B\cup R\right)\! &>&\!\nu \left( \left( B\cup Y\right) \cup \left( B\cup R\right) \right) \!+\nu \left( \left( B\cup Y\right) \cap \left( B\cup R\right) \right) \text{.}\end{array} $$

(23)

An analogous argument shows that the choice pair B,A ^′ requires local convexity of ν in the following sense

$$ \nu \left( B\cup Y\right) +\nu \left( B\cup R\right) <\nu \left( \left( B\cup Y\right) \cup \left( B\cup R\right) \right) +\nu \left( \left( B\cup Y\right) \cap \left( B\cup R\right) \right) \text{.} $$

(24)

Concavity, resp. convexity, of the non-additive probability measure ν is typically associated with optimistic, resp. pessimistic, ambiguity attitudes of a prospect theory decision maker (cf. Wakker 2001; Chapter 10.4.3 in Wakker 2010).

Appendix C: Risk and the independence axiom

Consider the limiting case of statistical learning in which the decision maker could observe an infinite number of drawings (with replacement) from the urn, i.e. an infinite amount of data generated by multivariate Bernoulli trials driven by the balls’ true proportions. According to standard models of statistical Bayesian learning, this decision maker will then learn, by Doob’s (1949) consistency theorem, with certainty the true probabilities of the events R, Y, and B.¹⁰

Denote this objective probability measure as π ^∗; e.g. in our experiment we stipulate that π ^∗(R)=π ^∗(Y)=0.2, π ^∗(B)=0.6 in accordance with the balls’ true proportions. In this risky decision situation with known probabilities, the prospects A, B, A ^′, and B ^′ can be reinterpreted as the following lotteries (=objective probability distributions) over the monetary outcomes x,y,z, respectively:

$$\begin{array}{@{}rcl@{}} \mathbf{A} &=&\left( 0.2,0.2,0.6\right) \text{,} \\ \mathbf{B} &=&\left( 0,1,0\right) \text{,} \\ \mathbf{A}^{\prime } &=&\left( 0.4,0,0.6\right) \text{,} \\ \mathbf{B}^{\prime } &=&\left( 0.2,0.8,0\right) \text{.} \end{array} $$

EU theory had been first axiomatized for preferences over lotteries by von Neuman and Morgenstern (1947). Key to their celebrated EU representation theorem is the independence axiom (IA) which implies that, for all lotteries L,L ^′,L ^″ and all λ∈(0,1),

$$ L\succsim L^{\prime }\Leftrightarrow \lambda \cdot L+\left( 1-\lambda \right) L^{\prime \prime }\succsim \lambda \cdot L^{\prime }+\left( 1-\lambda \right) L^{\prime \prime }\text{.} $$

(25)

Focus on the choice pair A,B ^′ and observe that it reveals a violation of the IA. To see this denote by δ _k , k∈{x,y,z}, the degenerate lottery that gives the monetary amount k with probability one. Let λ=π ^∗(R) and (1−λ)⋅μ=π ^∗(Y) and observe that (25) implies for the choice pair A,B ^′ that¹¹

$$\begin{array}{@{}rcl@{}} \mathbf{A\succ B} &\Leftrightarrow & \end{array} $$

(26)

$$\begin{array}{@{}rcl@{}} \lambda \cdot \delta_{y}+\left( 1-\lambda \right) \cdot \left[ \mu \cdot \delta_{x}+\left( 1-\mu \right) \cdot \delta_{z}\right] &>&\lambda \cdot \delta_{y}+\left( 1-\lambda \right) \cdot \delta_{y}\Leftrightarrow \end{array} $$

(27)

$$\begin{array}{@{}rcl@{}} \mu \cdot \delta_{x}+\left( 1-\mu \right) \cdot \delta_{z} &>&\delta_{y} \text{.} \end{array} $$

(28)

as well as

$$\begin{array}{@{}rcl@{}} \mathbf{B}^{\prime } &\mathbf{\succ }&\mathbf{A}^{\prime }\Leftrightarrow \end{array} $$

(29)

$$\begin{array}{@{}rcl@{}} \lambda \cdot \delta_{x}+\left( 1-\lambda \right) \cdot \delta_{y} &>&\lambda \cdot \delta_{x}+\left( 1-\lambda \right) \cdot \left[ \mu \cdot \delta_{x}+\left( 1-\mu \right) \cdot \delta_{z}\right] \Leftrightarrow \end{array} $$

(30)

$$\begin{array}{@{}rcl@{}} \delta_{y} &>&\mu \cdot \delta_{x}+\left( 1-\mu \right) \cdot \delta_{z} \text{.} \end{array} $$

(31)

Since (28) and (31) constitute a contradiction, the revealed preferences A,B ^′ violate the IA.

More generally, in a decision situation with known probabilities the choice pairs (1) are consistent with the IA, whereas the choice pairs (2) violate the IA.

Appendix D: Caveat

Whereas the choice pairs (2) unambiguously reveal a violation of the STP for the control group, the situation is slightly different for the test group whose members observe the drawing of one ball after each choice. To see this consider a decision maker of the test group and denote by I _n the information he has received by observing n drawings. If this decision maker chooses, e.g. A after n and B ^′ after n+1 drawings, these revealed choices could be rationalized as EU consistent choices as follows:

$$\begin{array}{@{}rcl@{}} \mathbf{A}\mathbf{\succ B} &\Leftrightarrow & \end{array} $$

(32)

$$\begin{array}{@{}rcl@{}} u\left( x\right) \cdot \pi \left( Y\mid I_{n}\right) +u\left( z\right) \cdot \pi \left( B\mid I_{n}\right) &>&u\left( y\right) \cdot \left( 1-\pi \left( R\mid I_{n}\right) \right) \end{array} $$

(33)

and

$$\begin{array}{@{}rcl@{}} \mathbf{B}^{\prime } &\succ &\mathbf{A}^{\prime }\Leftrightarrow \end{array} $$

(34)

$$\begin{array}{@{}rcl@{}} u\left( y\right) \cdot \left( 1-\pi \left( R\mid I_{n+1}\right) \right) &>&u\left( x\right) \cdot \pi \left( Y\mid I_{n+1}\right) +u\left( z\right) \cdot \pi \left(B\mid I_{n+1}\right) \text{.} \end{array} $$

(35)

Whereas the two inequalities (33) and (35) cannot simultaneously hold if π(⋅∣I _n ) and π(⋅∣I _n+1) are sufficiently similar probability measures, it is possible that (33) and (35) are satisfied for appropriately chosen values of π(⋅∣I _n ) and π(⋅∣I _n+1). In that case, the choices A,B ^′ would not reveal a violation of the STP but rather the different perception of the urn’s uncertainty by an EU decision maker before versus after he updates his additive belief on the n+1th observation.

We could have easily avoided this ambiguity in the interpretation of choice pairs (2) for the test group, if we had allowed for statistical learning not within but only after each question pair was answered. However, when designing the experiment, our concern was to give the subjects no hint that they were actually answering 15 well-structured question pairs rather than 30 similar questions. Although we feared that a detection of this question pair structure might eventually result in some answering bias, we assumed that any difference between π(⋅∣I _n ) and π(⋅∣I _n+1) would be negligibly small (in the sense of: How great can the impact of a single observation be?).

In the subsequent interpretation of the data, we therefore assume that π(⋅∣I _n ) and π(⋅∣I _n+1) are indeed sufficiently close so that choice pairs (2) cannot be explained by EU consistent decision making but rather by revealed violations of the STP. If this assumption was violated, however, we would observe, by Doob’s (1949) consistency theorem, that π(⋅∣I _n ) and π(⋅∣I _n+1) become more and more similar with increasing n. Applied to the experiment, this means that although revealed choices (2) in the first rounds of the experiment (small n) might be EU consistent, revealed choices (2) in the later rounds of the experiment (large n) would rather indicate a violation of the STP. Consequently, an EU consistent decision maker who expressed choice pairs (2) in the beginning of the experiment would eventually switch to choice pairs (1) in the later rounds of the experiment when his conditional probability measures converge through statistical learning in accordance with Doob’s (1949) consistency theorem.

To summarize the caveat: If our maintained assumption that choice pairs (2) always reveal violations of the STP for the test group was not correct, we might observe a decline in the number of choice pairs (2) which is not caused by a decline of violations of the STP.