Axiomatizing bounded rationality: the priority heuristic

Abstract

This paper presents an axiomatic framework for the priority heuristic, a model of bounded rationality in Selten’s (in: Gigerenzer and Selten (eds.) Bounded rationality: the adaptive toolbox, 2001) spirit of using empirical evidence on heuristics. The priority heuristic predicts actual human choices between risky gambles well. It implies violations of expected utility theory such as common consequence effects, common ratio effects, the fourfold pattern of risk taking and the reflection effect. We present an axiomatization of a parameterized version of the heuristic which generalizes the heuristic in order to account for individual differences and inconsistencies. The axiomatization uses semiorders (Luce, Econometrica 24:178–191, 1956), which have an intransitive indifference part and a transitive strict preference component. The axiomatization suggests new testable predictions of the priority heuristic and makes it easier for theorists to study the relation between heuristics and other axiomatic theories such as cumulative prospect theory.

Appendix

1.1 Proof of Theorem 1

1.1.1 Sufficiency

Statement 1 and 2: Define $\delta _P$ by Statement 1. Axioms 6 and 7 insure the existence of the supremum, and by Axioms 6(ii) and 6(iii), $\delta _P >0$. By the definition of $\delta _P$ and Axiom 7, Statement 2 follows.

Statement 3: By Axioms 2 and 6, and Theorem 16.7 of Suppes et al. (1989), there exists a real-valued function $\phi _P$ on $P$ such that $p_1 W(R_P) p_2$ iff $\phi _P(p_1) > \phi _P(p_2)$, and with the asserted uniqueness properties.

Statement 4: By Axiom 4 and Theorem 2.1 of Krantz et al. (1971), there exists a real-valued function $\phi _Y$ on $Y$ such that $y_1 R_Y y_2$ iff $\phi _Y(y_1) \ge \phi _Y(y_2)$, and with the asserted uniqueness properties.

Statement 5: Statement 4 says that $\phi _Y$ preserves $R_Y$. By the definition of $R_Y$, it is identical to $\succcurlyeq$ when $\succcurlyeq$ is applied to $P(p_3, p_4) \times Y$ and restricted to $Y$. So, $\phi _Y$ also preserves the order $\succcurlyeq$ when it is applied to $P(p_3, p_4) \times Y$ and restricted to $Y$. By Axiom 6 (ii), there are successive indifference intervals on $P$ with nontrivial regions of overlap. Forcing the local scales to agree yields a global scale on $P \times Y$. The restriction of this scale to $Y$, $\phi _Y$ preserves $R_Y$ as well. Statement 5 follows from this, together with the other four statements and the whole construction.

1.1.2 Necessity of Axioms 1–5

Axiom 1

The reflexivity and completeness of $\succcurlyeq$ follow immediately from Statement 5. To show independence of the first attribute from the second, consider a $y_1$ in $Y$ and assume $(p_1, y_1) \succcurlyeq (p_2, y_1)$. By Statement 5, this means $\phi _P (p_1) > \phi _P(p_2) +\delta _P (p_2)$, which in turn means that $(p_1, y_2) \succcurlyeq (p_2, y_2)$ for any $y_2$ in $Y$. To show independence of the second attribute from the first, consider a $p_1$ in $P$ and assume $(p_1, y_1) \succcurlyeq (p_1, y_2)$. By Statement 5, this means that $\phi _Y (y_1) > \phi _Y(y_2)$, which in turn means that $(p_2, y_1) \succcurlyeq (p_2, y_2)$ for any $p_2$ in $P$.

Axiom 2

Part (i) of the definition of a semiorder follows immediately from Statement 2.

For Part (ii) of the definition, we assume $p_1 R_P p_2$, $p_3 R_P p_4$ and show that if also not $p_1 R_P p_4$, then $p_3 R_P p_2$. By Statement 2, $p_1 R_P p_2$ implies $\phi _P(p_2)+ \delta _P (p_2) < \phi _P(p_1)$, and not $p_1 R_P p_4$ implies $\phi _P(p_1)\le \phi _P(p_4)+ \delta _P (p_4)$. Thus, also $\phi _P(p_2)+ \delta _P (p_2) < \phi _P(p_4)+ \delta _P (p_4)$. This, together with $\phi _P(p_4)+ \delta _P (p_4)< \phi _P(p_3)$ (which holds from $p_3 R_P p_4$ and Statement 2), means that $\phi _P(p_2)+ \delta _P (p_2) < \phi _P(p_3)$, or, by Statement 2, $p_3 R_P p_2$.

For Part (iii) of the definition of a semiorder, we assume $p_1 P_P p_2$ and $p_2 R_P p_3$, and considering a $p_4$ in $P$, we show that either $p_4 R_P p_3$ or $p_1 R_P p_4$. Specifically, we show that, if (a) $\phi _P(p_4) \ge \phi _P(p_2)$, then $p_4 R_P p_3$, and if (b) $\phi _P(p_4) < \phi _P(p_2)$, then $p_1 R_P p_4$.

For (a), $p_2 R_P p_3$ implies, by Statement 2, that $\phi _P(p_2) > \phi _P(p_3) + \delta _P(p_3)$. Together with $\phi _P(p_4) \ge \phi _P(p_2)$, this means $\phi _P(p_4) > \phi _P(p_3) + \delta _P(p_3)$, or, by Statement 2, $p_4 R_P p_3$.

For (b), we first show that $p_1 R_P p_4$ holds if additionally $\phi _P(p_4)+ \delta _P(p_4) \le \phi _P(p_2) + \delta _P(p_2)$. This, together with $\phi _P(p_2)+ \delta _P(p_2) < \phi _P(p_1)$ (by $p_1 R_P p_2$ and Statement 2), means that $\phi _P(p_4)+ \delta _P(p_4) < \phi _P(p_1)$, or, by Statement 2, $p_1 R_P p_4$ as required.

To complete the argument, we show by contradiction that $\phi _P(p_4)+ \delta _P(p_4) \le \phi _P(p_2) + \delta _P(p_2)$. Suppose $\phi _P(p_4)+ \delta _P(p_4) > \phi _P(p_2) + \delta _P(p_2)$. Then it is possible to find a $p_5$ in $P$ such that: $\phi _P(p_4)+ \delta _P(p_4) =\phi _P(p_5) > \phi _P(p_2) + \delta _P(p_2)$. By Statement 2, $\phi _P(p_5) > \phi _P(p_2) + \delta _P(p_2)$ implies $p_5 R_P p_2$.

By Statement 2, $\phi _P(p_4)+\delta _P(p_4)=\phi _P(p_5)$ implies that not $p_5 R_P p_4$. Also, by Statement 1,$\phi _P(p_4)+\delta _P(p_4)=\phi _P(p_5)$ implies that $\phi _P(p_4)<\phi _P(p_5)<\phi _P(p_5) +\delta _P(p_5)$. By Statement 2, this implies that not $p_4 R_P p_5$. Together, not $p_5 R_P p_4$ and not $p_4 R_P p_5$ imply that $p_5 I(R_P) p_4$.

By the assumption of (b), $\phi _P(p_4)<\phi _P(p_2)$ and by Statement 1, $\phi _P(p_4)<\phi _P(p_2)+\delta _P(p_2)$. By Statement 2 this implies that not $p_4 R_P p_2$. Furthermore, from $\phi _P(p_4)+\delta _P(p_4)>\phi _P(p_2)$, which we assumed for contradiction, it follows that not $p_2 R_P p_4$. From not $p_4 R_P p_2$ and not $p_2 R_P p_4$ it follows that $p_4 I(R_P) p_2$.

Having established $p_5 I(R_P) p_4$, $p_4 I(R_P) p_2$ and $p_5 R_P p_2$, by the definition of weak preference $p_4 W(R_P) p_2$.

By Statement 3, $p_4 W(R_P) p_2$ implies $\phi _P(p_4) > \phi _P (p_2)$ which is inconsistent with the assumption of (b), $\phi _P(p_4)< \phi _P(p_2)$. Whence, $\phi _P(p_4)+ \delta _P(p_4) \le \phi _P(p_2) + \delta _P(p_2)$ as required.

Axiom 3

By Statement 5, $\phi _P$ preserves the order $\succcurlyeq _P$ and by Statement 3, $\phi _P$ preserves the order $W(R_P)$, so $\succcurlyeq _P$ and $W(R_P)$ are identical.

Axiom 4

By Statement 4 and Theorem 2.1 of Krantz et al. (1971), Axiom 4 follows.

Axiom 5

By Statement 5, $\phi _Y$ preserves the order $\succcurlyeq _Y$ and by Statement 4, $\phi _Y$ also preserves $R_Y$, so $\succcurlyeq _Y$ and $R_Y$ are identical.