Probabilistic independence axiom

Abstract

One of the most well-known theories of decision making under risk is expected utility theory based on the independence axiom. The independence axiom postulates that decision maker’s preferences between two lotteries are not affected by mixing both lotteries with the same third lottery (in identical proportions). The probabilistic independence axiom (also known as the cancelation axiom) extends this classic independence axiom to situations when a decision maker chooses in a probabilistic manner (i.e., she does not necessarily prefer the same choice alternative when repeatedly presented with the same choice set). Probabilistic choice may occur for a variety of reasons such as unobserved attributes of choice alternatives, imprecision of preferences, random errors/noise in decisions. According to probabilistic independence axiom, the probability that a decision maker chooses one lottery over another does not change when both lotteries are mixed with the same third lottery (in identical proportions). This paper presents a model of probabilistic binary choice under risk based on this probabilistic independence axiom. The presented model generalizes an incremental expected utility advantage model of Fishburn (Int Econ Rev 19(3):633–646, 1978) and stronger utility model of Blavatskyy (Theory Decis 76(2):265–286, 2014).

Appendix

Proof of Proposition 1

Consider any two distinct lotteries $L\left( {p_{ 1} , \ldots ,p_{n} } \right) \in {\mathcal{L}}{\text{ and}}\;L^{{\prime }} \left( {q_{ 1} , \ldots ,q_{n} } \right) \in {\mathcal{L}}$ such that p_i≥ q_i for all i ∊ {1,…,k} and q_j> p_j for all j ∊ {k + 1,…,n} for some k ∊ {1,…,n − 1}.

First, lottery L′(q₁,…,q_n) is a reduced-form of a compound lottery $q_{ 1} \varvec{x}_{{\mathbf{1}}} \, + \,\left( { 1 - q_{ 1} } \right)L_{1}^{{\prime }} ,$ where x₁ denotes a degenerate lottery that yields outcome x₁ for sure and $L_{1}^{{\prime }}$ denotes lottery

$$\left( {0,\frac{{q_{2} }}{{1 - q_{1} }}, \ldots ,\frac{{q_{n} }}{{1 - q_{1} }}} \right)$$

Similarly, lottery L(p₁,…,p_n) is a reduced-form of a compound lottery $q_{ 1} \varvec{x}_{{\mathbf{1}}} \, + \,\left( { 1- q_{ 1} } \right)L_{ 1} ,$ where L₁ denotes lottery

$$\left( {\frac{{p_{1} - q_{1} }}{{1 - q_{1} }},\frac{{p_{2} }}{{1 - q_{1} }}, \ldots ,\frac{{p_{n} }}{{1 - q_{1} }}} \right)$$

If Axiom 1 holds then $P\left( {L,L^{{\prime }} } \right)\, \equiv \,P\left( {q_{ 1} x_{{\mathbf{1}}} \, + \,\left( { 1- q_{ 1} } \right)L_{ 1} ,q_{ 1} x_{{\mathbf{1}}} \, + \,\left( { 1- q_{ 1} } \right)L_{1}^{{\prime }} } \right)\, = \,P\left( {L_{ 1} ,L_{1}^{{\prime }} } \right).$

Second, $L_{1}^{{\prime }}$ is a reduced-form of a compound lottery $\left[ {q_{ 2} /\left( { 1 - q_{ 1} } \right)} \right]\varvec{x}_{{\mathbf{2}}} \, + \,\left[ { 1 - q_{ 2} /\left( { 1 - q_{ 1} } \right)} \right]L_{2}^{{\prime }} ,$ where x₂ denotes a degenerate lottery that yields outcome x₂ for sure and $L_{2}^{{\prime }}$ denotes lottery

$$\left( {0,0,\frac{{q_{3} }}{{1 - q_{1} - q_{2} }}, \ldots ,\frac{{q_{n} }}{{1 - q_{1} - q_{2} }}} \right)$$

Similarly, lottery L₁ is a reduced-form of a compound lottery $\left[ {q_{ 2} /\left( { 1 - q_{ 1} } \right)} \right]\varvec{x}_{{\mathbf{2}}} \, + \,\left[ { 1 - q_{ 2} /\left( { 1 - q_{ 1} } \right)} \right]L_{2} ,$ where L₂ denotes lottery

$$\left( {\frac{{p_{1} - q_{1} }}{{1 - q_{1} - q_{2} }},\frac{{p_{2} - q_{2} }}{{1 - q_{1} - q_{2} }},\frac{{p_{3} }}{{1 - q_{1} - q_{2} }}, \ldots ,\frac{{p_{n} }}{{1 - q_{1} - q_{2} }}} \right)$$

If Axiom 1 holds then $P\left( {L_{ 1} ,L_{1}^{{\prime }} } \right)\, \equiv \,P\left( {\left[ {q_{ 2} /\left( { 1- q_{ 1} } \right)} \right]x_{{\mathbf{2}}} \, + \,\left[ { 1- q_{ 2} /\left( { 1- q_{ 1} } \right)} \right]L_{ 2} ,\left[ {q_{ 2} /\left( { 1- q_{ 1} } \right)} \right]x_{{\mathbf{2}}} \, + \,\left[ { 1- q_{ 2} /\left( { 1- q_{ 1} } \right)} \right]L_{2}^{{\prime }} } \right)\, = \,P\left( {L_{ 2} ,L_{2}^{{\prime }} } \right).$

Iterating such application of Axiom 1 for the first k outcomes we obtain that $P\left( {L,L^{{\prime }} } \right)\, = \,P\left( {L_{k} ,L_{k}^{{\prime }} } \right),$ where L_k denotes lottery

$$\left( {\frac{{p_{1} - q_{1} }}{{1 - \mathop \sum \nolimits_{i = 1}^{k} q_{i} }}, \ldots ,\frac{{p_{k} - q_{k} }}{{1 - \mathop \sum \nolimits_{i = 1}^{k} q_{i} }},\frac{{p_{k + 1} }}{{1 - \mathop \sum \nolimits_{i = 1}^{k} q_{i} }}, \ldots ,\frac{{p_{n} }}{{1 - \mathop \sum \nolimits_{i = 1}^{k} q_{i} }}} \right)$$

and $L_{k}^{{\prime }}$ denotes lottery

$$\left( {\underbrace {{0, \ldots ,0}}_{k},\frac{{q_{{k + 1}} }}{{1 - \sum\nolimits_{{i = 1}}^{k} {q_{i} } }}, \ldots ,\frac{{q_{n} }}{{1 - \sum\nolimits_{{i = 1}}^{k} {q_{i} } }}} \right)$$

Lottery L_k is a reduced form of a compound lottery

$$L_{k} \equiv \frac{{p_{k + 1} }}{{1 - \mathop \sum \nolimits_{i = 1}^{k} q_{i} }}\varvec{x}_{{\varvec{k} + 1}} + \left( {1 - \frac{{p_{k + 1} }}{{1 - \mathop \sum \nolimits_{i = 1}^{k} q_{i} }}} \right)L_{k + 1}$$

where x_k+1 denotes a degenerate lottery that yields outcome x_k+1 for sure and L_k+1 denotes lottery

$$\left( {\frac{{p_{1} - q_{1} }}{{1 - \mathop \sum \nolimits_{i = 1}^{k} q_{i} - p_{k + 1} }}, \ldots ,\frac{{p_{k} - q_{k} }}{{1 - \mathop \sum \nolimits_{i = 1}^{k} q_{i} - p_{k + 1} }},0,\frac{{p_{k + 2} }}{{1 - \mathop \sum \nolimits_{i = 1}^{k} q_{i} - p_{k + 1} }}, \ldots ,\frac{{p_{n} }}{{1 - \mathop \sum \nolimits_{i = 1}^{k} q_{i} - p_{k + 1} }}} \right)$$

Similarly, lottery $L_{k}^{{\prime }}$ is a reduced form of a compound lottery

$$L'_{k} \equiv \frac{{p_{k + 1} }}{{1 - \mathop \sum \nolimits_{i = 1}^{k} q_{i} }}\varvec{x}_{{\varvec{k} + 1}} + \left( {1 - \frac{{p_{k + 1} }}{{1 - \mathop \sum \nolimits_{i = 1}^{k} q_{i} }}} \right)L'_{k + 1}$$

where $L_{k + 1}^{{\prime }}$ denotes lottery

$$\left( {\underbrace {{0, \ldots ,0}}_{k},\frac{{q_{{k + 1}} - p_{{k + 1}} }}{{1 - \sum\nolimits_{{i = 1}}^{k} {q_{i} } - p_{{k + 1}} }},,\frac{{q_{{k + 2}} }}{{1 - \sum\nolimits_{{i = 1}}^{k} {q_{i} } - p_{{k + 1}} }} \ldots ,\frac{{q_{n} }}{{1 - \sum\nolimits_{{i = 1}}^{k} {q_{i} } - p_{{k + 1}} }}} \right)$$

Axiom 1 then implies $P\left( {L_{k} ,L_{k}^{{\prime }} } \right)\, = \,P\left( {L_{k + 1} ,L_{k + 1}^{{\prime }} } \right).$

Iterating such application of Axiom 1 for the remaining n − k − 1 outcomes we obtain that $P\left( {L,L^{{\prime }} } \right)\, = \,P\left( {L_{n} ,L_{n}^{{\prime }} } \right)$ where L_n denotes lottery

$$\left( {\frac{{p_{1} - q_{1} }}{{\sum\nolimits_{{i = 1}}^{k} {\left( {p_{i} - q_{i} } \right)} }}, \ldots ,\frac{{p_{k} - q_{k} }}{{\sum\nolimits_{{i = 1}}^{k} {\left( {p_{i} - q_{i} } \right)} }},\underbrace {{0, \ldots ,0}}_{{n - k}},} \right)$$

and $L_{n}^{{\prime }}$ denotes lottery

$$\left( {\underbrace {{0, \ldots ,0}}_{k},\frac{{q_{{k + 1}} - p_{{k + 1}} }}{{\sum\nolimits_{{j = k + 1}}^{n} {\left( {q_{j} - p_{j} } \right)} }}, \ldots ,\frac{{q_{n} - p_{n} }}{{\sum\nolimits_{{j = k + 1}}^{n} {\left( {q_{j} - p_{j} } \right)} }}} \right)$$

The last non-zero probability in lottery L_n is one minus the sum of the first k − 1 probabilities:

$$\frac{{p_{k} - q_{k} }}{{\mathop \sum \nolimits_{i = 1}^{k} \left( {p_{i} - q_{i} } \right)}} = 1 - \mathop \sum \limits_{r = 1}^{k - 1} \frac{{p_{r} - q_{r} }}{{\mathop \sum \nolimits_{i = 1}^{k} \left( {p_{i} - q_{i} } \right)}}$$

The last probability in lottery $L_{n}^{{\prime }}$ is one minus the sum of the proceeding n − k − 1 probabilities:

$$\frac{{q_{n} - p_{n} }}{{\mathop \sum \nolimits_{j = k + 1}^{n} \left( {q_{j} - p_{j} } \right)}} = 1 - \mathop \sum \limits_{s = k + 1}^{n - 1} \frac{{q_{s} - p_{s} }}{{\mathop \sum \nolimits_{j = k + 1}^{n} \left( {q_{j} - p_{j} } \right)}}$$

Thus, P(L,L′) can be written as function (1) of n − 2 probabilities, where function $F:\left[ {0,1} \right]^{n - 2} \to \left[ {0,1} \right]$ denotes the probability that a decision maker chooses lottery L_n over lottery $L_{n}^{{\prime }} .$****