# Regret theory, Allais’ Paradox, and Savage’s omelet

V.G. Bardakhchyan1,2) and A.E. Allahverdyan1,2) 1)Alikhanian National
Laboratory (Yerevan Physics Institute), Alikhanian Brothers Street 2, Yerevan
0036, Armenia,
2)Yerevan State University, 1 A. Manoogian street, Yerevan 0025, Armenia

###### Abstract

We study a sufficiently general regret criterion for choosing between two
probabilistic lotteries. For independent lotteries, the criterion is
consistent with stochastic dominance and can be made transitive by a unique
choice of the regret function. Together with additional (and intuitively
meaningful) super-additivity property, the regret criterion resolves the
Allais’ paradox including the cases were the paradox disappears, and the
choices agree with the expected utility. This super-additivity property is
also employed for establishing consistency between regret and stochastic
dominance for dependent lotteries. Furthermore, we demonstrate how the regret
criterion can be used in Savage’s omelet, a classical decision problem in
which the lottery outcomes are not fully resolved. The expected utility cannot
be used in such situations, as it discards important aspects of lotteries.

Keywords: Regret theory, Allais’ paradox, stochastic dominance, transitive
regret.

JEL Classification: D81.

##  I Introduction

The history of expected utility theory (EUT) started with Bernoulli’s work
resolving the St. Petersburg paradox stearns . Several axiomatic schemes for
EUT are known luce ; savage . Currently, EUT has applications in a wide range
of fields, including economics Hey_theories , psychology baron , evolutionary
game theory sigmund , and general artificial intelligence everitt .

EUT shows how to choose between two lotteries luce ; savage ; Hey_theories :

$\displaystyle(x,p)=\begin{pmatrix}x_{1}&x_{2}&...&x_{n}\\\
p_{1}&p_{2}&...&p_{n}\end{pmatrix},\qquad(y,q)=\begin{pmatrix}y_{1}&y_{2}&...&y_{n}\\\
q_{1}&q_{2}&...&q_{n}\end{pmatrix},$ (1)
$\displaystyle{\sum}_{k=1}^{n}p_{k}={\sum}_{k=1}^{n}q_{k}=1,$ (2)

where $(p_{1},...,p_{n})$ and $(q_{1},...,q_{n})$ are (resp.) the
probabilities of monetary outcomes $(x_{1},...,x_{n})$ and $(y_{1},...,y_{n})$
within each lottery. EUT proposes the following functional for each lottery
luce ; savage ; Hey_theories :

$V(x,p)={\sum}_{i=1}^{n}u(x_{i})p_{i},$ (3)

where $u(x_{i})$ is the utility of the monetary value $x_{i}$. EUT recommends
choosing in (1) the first lottery, if $V(x,p)>V(y,q)$.

Experiments revealed problems with EUT and its axiomatic foundations. In
particular, several classic experiments cannot be explained by EU for any
choice of the utility function $u(.)$ in (3) allais ; Machina . People
generally choose in contradiction to EUT, violating the independence axiom,
one of four axioms of the von Neumann-Morgenstern formulation of EUT luce .
The most prominent example of this is Allais’s paradox allais , where each
human subject chooses between two lotteries. The prospect theory Prospect ;
Cumul_Prospect , and rank-dependent utility theory Quiggin ; wak  discarded
the independence axiom, and proposed functionals similar to $V(x,p)$ in (3),
where instead of probabilities $p_{i}$ one employs weights $\pi_{i}$ that
generally depend both on $(p_{1},...,p_{n})$ and $(x_{1},...,x_{n})$. Refs.
Machina ; Hey_theories ; baron  discuss these and other alternatives to EUT.

There are also other situations where EUT does not apply. EUT cannot be used
directly when the lottery outcome remains uncertain even after the lottery
choice has been made. A good example of this situation is the decision problem
known as Savage’s omelet savage . In our knowledge, this problem has never
been studied from a viewpoint of EUT’s inapplicability.

As we show below, both Allais’ paradox and Savage’s omelet can be resolved by
the regret theory (RT), which is one of the alternatives of EUT. The main
difference of RT compared to EUT is that RT does not operate with a value
functional for a single lottery. Instead it counter-factually compares two
lotteries. RT has an intuitive emotional appeal, and it is also related to
cognitive aspects of decision making bourgeois2010regret . RT was first
proposed by Savage in minimax form savage  [see tempered  for an update of
this approach], and later brought to its current form in Regret_Sugden ; Bell
; see Regret_main ; Machina  for a review. Ref. Regret_Sugden  extended the
regret to independent lotteries and noted its potential in explaining Allais’
paradox. Ref. Regret_Sugden  also analyzed transitivity, common ration effect,
and preference reversals. Functional forms involving two lotteries were given
axiomatic foundation in Fishburn_SSB . An axiomatic formulation of regret was
attempted in Reg_axioms .

This work has three purposes. First, we want to show how Allais’ paradox is
solved by a transitive and super-additive RT. People mentioned regret in the
context of Allais’ paradox [see e.g. baron ; Regret_Sugden ;
bourgeois2010regret ], but so far no systematic and complete solution of this
paradox was provided. Our solution is rather complete, because it also
predicts conditions under which the paradox does not hold. Both transitivity
and super-additivity have transparent meaning for regret theories in general.
We do clarify their applicability range. This is especially important for
transitivity, because generally regret theories do not lead to transitive
predictions Starmer .

Second, we prove that the transitive and super-additive regret theory is
consistent with the stochastic dominance criterion luce . Stochastic dominance
is a useful tool, but it does not apply to comparing any pair of lotteries.
The previous literature in this direction is mostly negative showing that
regret-based approaches violate first order stochastic dominance Quiggin_std ;
levy2017regret  111Ref. Quiggin_std  analyzed relations between RT and
stochastic dominance for a specific case. This analysis is based on more
general formulation of first order stochastic dominance that compares
cumulative distribution functions. Here we focus on the simplest version of
stochastic dominance.. Third, we demonstrate—using as an example Savage’s
omlet problem—that RT can recommend choosing between lotteries with not
resolved outcomes, a task which cannot be consistently addressed by EUT.

The paper is organized as follows. Section II is devoted to regret functional
for independent lotteries and some of its properties related to the expected
utility. In decision making theory the functional form is frequently derived
from axiomatic foundation. In contrast, here we first introduce the functional
considered, then derive its properties. Section III is devoted to Allais’
paradox and its relations to other concepts. Stochastic dominance abidance is
considered in section IV. Section V analyzes Savage’s omelet problem,
identifies an aspect that prevents the applicability of the expected utility
theory, and solves this problem via the regret. We summarize in the last
section.

##  II Regret and its features

###  II.1 Axioms of Expected Utility Theory (EUT)

We remind the four axioms of EUT (3)—completeness, transitivity, continuity,
independence—since they will motivate our further consideration. First of all
one introduces a preference relation $\succeq$, and indifference relation
$\sim$ between the lotteries (1), where $\sim$ means that both $\succeq$ and
$\preceq$ hold. When comparing two lotteries in (1) we sometimes assume
(without loss of generality) the same outcomes: $\\{x_{k}=y_{k}\\}_{k=1}^{n}$.
If they are initially different, we can introduce suitable zero-probability
events and make them identical.

1. The completeness axiom states that any pair of lotteries in (1) can be compared:

$\displaystyle(x,p)\succeq(x,q)\quad{\rm or}\quad(x,q)\succeq(x,p)\quad{\rm
or}\quad(x,q)\sim(x,p),$ (4)

where $(x,p)\succeq(x,q)$ means that lottery $(x,q)$ is not preferred to
$(x,p)$.

2. The transitivity axiom states:

$\displaystyle(x,p)\succeq(x,q)\succeq(x,r)\quad{\rm
means}\quad(x,p)\succeq(x,r),$ (5)

3. The continuity axiom states for any three lotteries

$\displaystyle(x,p)\succeq(x,q)\succeq(x,r)\quad{\rm
implies}\quad(x,q)\sim(x,\alpha p+(1-\alpha)r),$ (6)

for some $\alpha\in[0,1]$. This axiom implies continuity of the value function
to be deduced from the four axioms.

4. The independence axiom—also known independence of irrelevant alternatives or the sure-thing principle—claims that combining each of two lotteries with any fixed one will not alter the preferences baron ; jeffrey :

$\displaystyle(x,p)\succeq(x,q)\quad{\rm means}\quad(x,\alpha
p+(1-\alpha)r)\succeq(x,\alpha q+(1-\alpha)r),$ (7)

where the irrelevant alternative is $(x,r)$. Eq. (7) is among the most
controversial axioms in decision theory and has triggered many debates baron ;
jeffrey ; see in this context also Appendix B, where we explain why
specifically the meaning of (7) can be ambiguous. Ref. pearl  briefly reviews
its current status with counter-examples. Experimental studies showed
violations of (7), with some concerns on whether these violations are
systematic baron .

###  II.2 Definition of regret

The regret defines a counterfactual outcome-wise comparison between the
lotteries (1) using certain ideas of EUT. Hence for particular cases it would
coincide with the decision criterion of EUT. The utility function $u(x)$ is
assumed to exist beforehand and known to the decision-maker baron .

Assume that $(y,q)$ is chosen and its outcome $y_{j}$ is found. The decision-
maker compares this outcome with what would be found if $(x,p)$ would be taken
and defines:

$\displaystyle R(x,p;y_{j})\equiv{\sum}_{i=1}^{n}f(u(x_{i})-u(y_{j}))p_{i},$
(8)

where $u(x)$ is the utility function, and $f(x)$ is a function holding

$\displaystyle f(x\geq 0)\geq 0,\qquad f(x\leq 0)\leq 0,\qquad f(0)=0.$ (9)

In particular, $R(x,p;y_{j})>0$ (positive regret), if $x_{i}>y_{j}$.
Generally, $f(x)$ accounts for both regret and appreciation. We get a pure
regret (appreciation), if $f(x\leq 0)=0$ ($f(x\geq 0)=0$).

Since $(x,p)$ was not actually chosen, its outcomes are not known; hence the
averaging in (8). Moreover, once the decision-maker keeps on choosing $(y,q)$
and explores all its outcomes according to their probabilities, the average of
(8) reads:

$\displaystyle
R(x,p;y,q)\equiv{\sum}_{j=1}^{n}q_{j}R(x,p;y_{j})={\sum}_{i,j=1}^{n}f(u(x_{i})-u(y_{j}))p_{i}q_{j},$
(10)

where (10) already assumed that the events $(y_{j},x_{i})$ are independent,
i.e. their joint probability is $q_{j}p_{i}$. This additional information is
to be provided for unambiguous definition of lotteries in (1).

Note that (8, 10) are asymmetric with respect to the lotteries (1), because
$(y,q)$ is actually chosen, while $(x,p)$ is reasoned counter-factually given
this choice. The regret preference $\succeq_{\rm reg}$ is defined as
Regret_Sugden ; Bell ; Regret_main ; Machina

$\displaystyle(x,p)\succeq_{\rm reg}(y,q)\quad{\rm iff}\quad
R(y,q;x,p)-R(x,p;y,q)={\sum}_{i,j=1}^{n}g(u(y_{j})-u(x_{i}))p_{i}q_{j}\leq 0,$
(11)

where

$\displaystyle g(x)\equiv f(x)-f(-x),$ (12)

is anti-symmetric and monotonic:

$\displaystyle g(x)=-g(-x),$ (13) $\displaystyle g(x)\geq g(y)\quad{\rm
for}\quad x\geq y.$ (14)

The meaning of $R(y,q;x,p)-R(x,p;y,q)\leq 0$ is that $(x,p)$ is preferred if
its leads to a smaller average regret. For a particular case

$\displaystyle g(x)=ax,\qquad a>0,$ (15)

where $a$ is a constant, we revert from (11) to the expected utility. Note
that (15) is achieved for various functions $f(x)$; e.g. $f(x)=ax/2$ or
$f(x)=a\,{\rm max}[x,0]$.

The above definition generalizes for a non-trivial joint probability
$P(x_{i},y_{j})$ of $(x_{i},y_{j})$ with

$\displaystyle{\sum}_{i=1}^{n}P(x_{i},y_{j})=q_{j},\qquad{\sum}_{j=1}^{n}P(x_{i},y_{j})=p_{i}.$
(16)

Now $p_{i}$ in (8) should be replaced by conditional probability
$P(x_{i}|y_{j})$, which is reasonable for a counter-factual reasoning, and
instead of (8–11) we have

$\displaystyle
R(x,p;y_{j})\equiv{\sum}_{i=1}^{n}f(u(x_{i})-u(y_{j}))P(x_{i}|y_{j}),$ (17)
$\displaystyle
R(x,p;y,q)\equiv{\sum}_{j=1}^{n}q_{j}R(x,p;y_{j})={\sum}_{i,j=1}^{n}f(u(x_{i})-u(y_{j}))P(x_{i},y_{j}),$
(18) $\displaystyle(x,p)\succeq_{\rm reg}(y,q)\quad{\rm
iff}\quad{\sum}_{i,j=1}^{n}g(u(y_{j})-u(x_{i}))P(x_{i},y_{j})\leq 0.$ (19)

In particular, the outcomes in (16) can refer to the same states of nature
Reg_axioms ; luce ; jeffrey . This implies

$\displaystyle P(x_{i},y_{j})=p_{i}\delta_{ij},\quad i,j=1,...,n,$ (20)

where $\delta_{ij}$ is the Kroenecker delta, and where
$\\{p_{i}=q_{i}\\}_{i=1}^{n}$ are the probabilities for those unknown states
of nature; see section V for details.

###  II.3 Two propositions about regret

Note that for the regret preference relation (11) we can take lotteries (1) to
have the same outcomes, $x_{k}=y_{k}$, using the same argument as before (4).
Now the completeness axiom (4) obviously holds for $\succeq_{\rm reg}$. The
continuity axiom is valid as well.

Proposition 1. For the regret preference relation (11)

$\displaystyle(x,p)\succeq_{\rm reg}(x,q)\succeq_{\rm reg}(x,r)\quad{\rm
implies}\quad(x,q)\sim_{\rm reg}(x,\alpha p+(1-\alpha)r),$ (21)

for some $\alpha\in[0,1]$. Working out the last relation in (21) we find

$\displaystyle\alpha=B/(A+B)\in[0,1],$ (22) $\displaystyle
A={\sum}_{i,j=1}^{n}p_{i}q_{j}g(u(x_{i})-u(x_{j}))\geq 0,\qquad
B={\sum}_{i,j=1}^{n}r_{i}q_{j}g(u(x_{j})-u(x_{i}))\geq 0,$ (23)

where (23) follows from first and second relations in (21).

It is known that $\succeq_{\rm reg}$ violates transitivity for a general
choice of $f(x)$ Starmer . In particular, the transitivity is violated under
(20) Transitive_regret ; e.g. for the same states of nature. Transitivity
violation is not necessarily a drawback, since there are arguments for
involving non-transitive choices even in normative choices
Fishburn_nontransitive . Ref. Transitive_regret_ind  shows that for the most
general form of regret there exist models not violating transitivity.

Let us now provide a sufficiently complete solution for the transitivity of
$\succeq_{\rm reg}$. First, we show that $\succeq_{\rm reg}$ will be
transitive for a particular choice of $f(x)$ in (11). Define

$\displaystyle f(x)=b(a^{x}-1),$ (24)

where $a>0$ and $b>0$. Eq. (9) holds. Now $(x,p)\succeq_{\rm reg}(x,q)$
amounts to

$\displaystyle v(p)w(q)\geq v(q)w(p),$ (25) $\displaystyle
v(p)\equiv{\sum}_{i=1}^{n}a^{u(x_{i})}p_{i}>0,\qquad
w(q)\equiv{\sum}_{i=1}^{n}a^{-u(x_{i})}q_{i}>0.$ (26)

Eqs. (25, 26) imply that with the choice (24), $\succeq_{\rm reg}$ is
transitive. Fisburn’s theorem on transitivity Fishburn  shows that (24) is
also necessary for transitivity.

Proposition 2. The regret preference relation $\succeq_{\rm reg}$ given by
(11) preserves transitivity iff (24) holds.

Returning to (4–7) we see that only the independence axiom can be violated by
$\succeq_{\rm reg}$; see below for more details.

##  III Solving Allais’ paradox with regret

There was a great deal of attention focused on Allais’ paradox as one of the
major systematic violations of EUT baron ; allais ; Prospect ; Cumul_Prospect
; Machina ; schoemaker . Regret theory is mentioned in the context of Allais’s
paradox baron ; bourgeois2010regret ; Regret_Sugden , but no systematic
solution of the paradox via the regret theory was so far provided. We show
below that this solution can be achieved by respecting the transitivity and
that it does provide an important constraint on the form of $g(x)$ in (11,
12).

Consider the standard formulation of the Allais’ paradox baron ; allais . A
decision make is choosing between the following two lotteries [cf. (1)]:

${\rm I}\equiv\begin{pmatrix}1\\\ 1\end{pmatrix},\qquad{\rm
II}\equiv\begin{pmatrix}0&1&5\\\ 0.01&0.89&0.1\end{pmatrix},$ (27)

and then between

${\rm III}\equiv\begin{pmatrix}0&1\\\ 0.89&0.11\end{pmatrix},\qquad{\rm
IV}\equiv\begin{pmatrix}0&5\\\ 0.9&0.1\end{pmatrix},$ (28)

where the monetary outcomes in (27, 28) are normally given in millions of $.

There are 4 possible outcomes here: $({\rm I},{\rm III})$, $({\rm I},{\rm
IV})$, $({\rm II},{\rm III})$, $({\rm II},{\rm IV})$, where $({\rm I},{\rm
III})$ means choosing ${\rm I}$ in (27) and ${\rm III}$ in (28). Choosing
$({\rm I},{\rm III})$ or $({\rm II},{\rm IV})$ is consistent with the EUT;
e.g. $({\rm I},{\rm III})$ is achieved if $u(1)<u(5)$ and $u(1)\approx u(5)$.
In contrast, most of people take $({\rm I},{\rm IV})$ thereby violating the
expected utility theory (EUT) baron .

Applying preference relation (11) to the choice $({\rm I},{\rm IV})$, we will
find an important and intuitive condition for function $g(x)$. Now ${\rm
I}\succeq_{\rm reg}{\rm II}$ reads from (11):

$0.01\cdot g(u(0)-u(1))+0.1\cdot g(u(5)-u(1))<0.$ (29)

Since $g(x)$ is an increasing function [cf. (14)], (29) implies

$u(5)-u(1)<u(1)-u(0).$ (30)

Thus (30)—which can be realized with a concave function $u(x)$ and hence
relates to risk-aversion—is a necessary condition for (11) to explain Allais’
paradox. Likewise, demanding ${\rm IV}\succeq_{\rm reg}{\rm III}$ in (28) we
get

$0.089\cdot g(u(5)-u(0))-0.099\cdot g(u(1)-u(0))+0.011\cdot g(u(5)-u(1))>0$
(31)

Taking the difference of (31) and (29) we get

$-0.089\cdot g(u(5)-u(0))+0.089\cdot g(u(1)-u(0))+0.089\cdot g(u(5)-u(1))<0,$

yielding

$g(u(5)-u(0))>g(u(1)-u(0))+g(u(5)-u(1)).$ (32)

Now (32) is the second necessary condition for solving Allais’s paradox.
Taking (32) and (29) together is necessary and sufficient for solving the
paradox. It is intuitively clear what (32) means. The decision maker is more
impressed (i.e. experiences more regret) with the difference $u(5)-u(0)$, than
with this difference $u(5)-u(0)=u(1)-u(0)+u(5)-u(1)$ coming in two separate
pieces: $u(1)-u(0)$ and $u(5)-u(1)$. We rewrite (32) as a more general
condition:

$g(x+y)\geq g(x)+g(y),\quad x\geq 0,\quad y\geq 0,$ (33)

which is the super-additivity (in positive domain) for $g(x)$. Noting from
(13) that $g(0)=0$, we recall that any convex function $g(x)$ with $g(0)=0$ is
super-additive 222This fact should be known, but let us present its short
proof. First note that $g(tx)\leq tg(x)$ for $0<t<1$ due to $g(t(x)+(1-t)\cdot
0)\leq tg(x)+(1-t)g(0)=tg(x)$. Next,
$g(x)+g(y)=g\left((x+y)\frac{x}{x+y}\right)+g\left((x+y)\frac{y}{x+y}\right)\leq\frac{x}{x+y}g(x+y)+\frac{y}{x+y}g(x+y)=g(x+y)$..
A simple example of a function that is easily shown to be super-additive, but
is not convex is $g(x)=x\,e^{-x^{-2}}$ japan . Indeed, $\frac{{\rm
d}^{2}}{{\rm d}x^{2}}g(x)=2\,e^{-x^{-2}}\,x^{-5}(2-x^{2})$, i.e. $g(x)$ is
concave (convex) for $x>\sqrt{2}$ ($\sqrt{2}>x>0$) 333Ref. Reg_axioms
mentioned the super-additivity condition in the context of regret. Ref.
Regret_Sugden  employed convexity (concavity) features of regret functional,
but without any definite reason.. We formulate our results as follows.

Proposition 3. Allais’s paradox can be explained by regret, if and only if
function $g(x)$ in (11) is strongly super-additive for some values in positive
domain.

Example. We take the transitive regret and logarithmic utility [cf. (13, 24)]

$\displaystyle g(x)=\sinh\left(\frac{x}{\beta}\right),\qquad
u(x)=\ln\left(\frac{x}{\gamma}+1\right),$ (34)

where $\beta>0$ and $\gamma>0$ are positive parameters that characterize the
decision maker. Here $\gamma>0$ defines the threshold of the concave (risk-
averse) utility $u(x)$ ($u(0)=0$), because only for $\frac{x}{\gamma}\ll 1$ we
have $u(x)\simeq 0$. In a sense, $\gamma$ defines the initial money, since
only for $\frac{x}{\gamma}\gtrsim 1$ the decision maker will care about money.
Likewise, $\beta$ has a similar meaning of threshold, but for the regret
function: if $\frac{x}{\beta}\ll 1$, then
$g(x)=\sinh(\frac{x}{\beta})\simeq\frac{x}{\beta}$ is effectively in the
regime EUT.

Now $g(x)$ in (34) holds super-additivity condition (33), since $\sinh(0)=0$
and $\frac{{\rm d}^{2}}{{\rm d}x^{2}}\sinh(x)=\sinh(x)\geq 0$ for $x\geq 0$;
hence (32) holds. For solving Allais’ paradox we need to look at condition
(29), which from (34) amounts to

$\displaystyle\gamma<\zeta(\beta),$ (35)
$\displaystyle\zeta(\beta\to\infty)=5^{-10},\quad\zeta(1)=0.021,\quad\zeta(\beta\to
0)=1/3.$ (36)

Hence $\zeta(\beta)$ changes from $5^{-10}$ to $1/3$, when $\beta$ moves from
$\infty$ to $0$. Let us focus on $\gamma<0.021$ in (36). We know that (27, 28)
are to be given in millions of $. Hence we multiply both $x$ and $\gamma$ in
$u(x)=\ln(\frac{x}{\gamma}+1)$ by $10^{6}$, and reach the following
conclusion: starting from the initial money $\geq 21000$ $ the decision maker
will behave according to the expected utility and choose lotteries $({\rm
II},{\rm IV})$ in (27, 28). The interpretation of the other two values of
$\zeta(\beta)$ in (36) is similar. Note in this context that $5^{-10}$ is
equivalent to $5^{-10}\times 10^{8}\simeq 10$ cents.

It is reported that with smaller outcomes—not millions of $ in (27,
28)—Allais’ paradox need not hold Small_outcomes ; camerer1989experimental ;
blavatskyy2015now . Other authors note that when shifting all outcomes in (27,
28) with the same substantial positive amount, Allais’ paradox will not hold
(aversion of ”0” outcome) 0_aversion . The scheme given by (34) handles both
experimental results.

Remark 1. The super-additivity (33) of $g(x)$ (and its ensuing relations with
convexity) does not relate to risk-aversion and risk-seeking, as defined via
utility $u(x)$. To understand this, compare the following two lotteries:

$\begin{pmatrix}x\\\ 1\end{pmatrix}\quad{\rm
and}\quad\begin{pmatrix}x-\epsilon&x+\epsilon\\\
0.5&0.5\end{pmatrix},\qquad\epsilon>0.$ (37)

Now the first (certain) lottery is regret-preferable compared with the second
(uncertain) lottery if $g(u(x)-u(x-\epsilon))>-g(u(x)-u(x+\epsilon))$, which
is achieved due to a monotonically increasing $g(x)$, and concavity of $u(x)$;
i.e. the risk-aversion at the level of the utility. Likewise, the convexity of
$u(x)$ (risk-seeking utility) will lead to preferring the uncertain lottery.

Remark 2. Note that the regret is invariant with respect to $u(x)\to u(x)+a$,
where $a$ is arbitrary, but it is not invariant with respect to $u(x)\to
bu(x)$, where $b>0$; see e.g. the very example (34). After transformation
$u(x)\to bu(x)$, one can redefine $g_{b}(x)=g(bx)$ such that the regret stays
invariant. This redefinition respects transitivity and super-additivity of
$g(x)$.

Remark 3. Recall that the independence axiom (7) (or the axiom of irrelevant
alternatives) is the main axiom of EUT violated by the regret theory. Allais’
paradox can be reformulated in such a way that the presence of this axiom is
made obvious. To this end one writes (27, 28) as

$\displaystyle{\rm I}=\begin{pmatrix}1&1&1\\\ 0.01&0.1&0.89\end{pmatrix},$
$\displaystyle{\rm III}=\begin{pmatrix}1&1&0\\\ 0.01&0.1&0.89\end{pmatrix},$
(38) $\displaystyle{\rm II}=\begin{pmatrix}0&5&1\\\
0.01&0.1&0.89\end{pmatrix},$ $\displaystyle{\rm IV}=\begin{pmatrix}0&5&0\\\
0.01&0.1&0.89\end{pmatrix}.$ (39)

We emphasize that ${\rm I}$ and ${\rm II}$ in (38, 39) (as well as ${\rm III}$
and ${\rm IV}$) refer to independent events.

It is seen that ${\rm I}$ and ${\rm II}$ have the common last column $({1\atop
0.89})$, while for ${\rm III}$ and ${\rm IV}$ the common last column is
$({0\atop 0.89})$. These last columns (i.e. the corresponding outcomes with
their probabilities) plays the role of independent alternatives. If they are
deemed to be irrelevant, e.g. $({1\atop 0.89})$ is irrelevant when deciding
between ${\rm I}$ and ${\rm II}$, then ${\rm I}$ becomes equivalent to ${\rm
III}$, and ${\rm II}$ is equivalent to ${\rm IV}$. Hence one takes either
$({\rm I},{\rm III})$ or $({\rm II},{\rm IV})$. Note that this reasoning is
more general than appealing directly to the axiom (7), since this mathematical
axiom does not specify the interpretation of the mixture model $\alpha
p+(1-\alpha)r$; see Appendix B for details.

If experimental subjects are presented Allais’ lotteries in the form (38, 39),
then majority of them behave according to EUT than for (27, 28) baron .
Naturally, for the regret (11) the difference between (38, 39) and (27, 28) is
absent. Hence these subjects did not use the regret theory in their decision
making.

##  IV Regret and stochastic dominance

For lotteries (1) with independent probabilities, a clear-cut definition of
superiority is provided by the stochastic dominance $\succeq_{\rm sto}$ luce .
Recall its definition: we assume 444This assumption of identical outcomes is
not necessary, since the stochastic dominance can be formulated more
generally. We do not focus on this general definition, since it is equivalent
to the situation, when the outcomes are made the same by increasing their
number via adding zero-probability events; cf. the discussion before (4).
that $x_{k}=y_{k}$ in (1) hold with

$\displaystyle x_{i}<x_{j}\quad{\rm for}\quad i<j.$ (40)

Now define luce

$\displaystyle(x,p)\succeq_{\rm sto}(x,q)\quad{\rm
iff}\quad{\sum}_{i=1}^{k}p_{i}\leq{\sum}_{i=1}^{k}q_{i}\quad{\rm for}\quad
k=1,..,n.$ (41)

Recall that the utility $u(x)$ in (11) is an increasing function of $x$.
Stochastic dominance does not depend on a specific form of the utility $u(x)$
in (11) provided that it is an increasing function of $x$, as we assume. This
is an advantage of stochastic dominance. Its weakness is that it clearly does
not apply to all lotteries, i.e. the completeness axiom (4) is violated.
Indeed, it is sufficient to violate (41) for one value of $k$, and this will
make $\succeq_{\rm sto}$ inapplicable. A related weakness is that its
applicability is not stable with respect to small variations of outcomes. To
see this, assume that (40, 41) hold and perturb $y_{1}=x_{1}\to
y_{1}^{\prime}<x_{1}$. Even a small variation of this type violates condition
(41) for $k=1$.

Regret and stochastic dominance do not contradict each other, as the following
proposition shows.

Proposition 4. $(x,p)\succeq_{\rm sto}(x,q)$ implies $(x,p)\succeq_{\rm
reg}(x,q)$ defined from (11). The proof is given in Appendix A.

Note that Proposition 4 does not require any specific feature of $g(x)$ apart
of (13, 14). However, it does require independent probabilities for the
lotteries, as implied by (11). Lotteries with independent probabilities have
vast but still limited range of applications. Even within the framework of
initially independent lotteries, one can envisage new dependent lotteries for
which the regret is given via (19). For dependent lotteries the relation
between regret and stochastic dominance is partially explained by the
following proposition.

Proposition 5. For the joint probability $P(x_{i},x_{j})$ given by (16), let
us define the marginal probabilities $\\{p_{i}\\}_{i=1}^{n}$ and
$\\{q_{j}\\}_{j=1}^{n}$, as well as deviation of $P(x_{i},x_{j})$ from
$p_{i}q_{j}$:

$\displaystyle p_{i}:={\sum}_{j=1}^{n}P(x_{i},x_{j}),\quad
q_{j}:={\sum}_{i=1}^{n}P(x_{i},x_{j}),$ (42)
$\displaystyle\theta_{i,j}:=P(x_{i},x_{j})-p_{i}q_{j},$ (43)
$\displaystyle{\sum}_{i=1}^{n}\theta_{i,j}={\sum}_{j=1}^{n}\theta_{i,j}=0,\qquad|\theta_{i,j}|\leq
p_{i}q_{j}.$ (44)

Then if $g(x)$ is super-additive on positive domain [see (33)] and if

$\displaystyle\theta_{i,j}\geq\theta_{j,i},\quad{\rm for}\quad i>j,$ (45)

one has that $(x,p)\succeq_{\rm sto}(x,q)$ defined via (40, 41) leads to
$(x,p)\succeq_{\rm reg}(x,q)$ in the sense of (19).

Thus the super-additivity of $g(x)$ plus condition (45) make the regret
consistent with the stochastic dominance. The proof of Proposition 5 is given
in Appendix C.

##  V Savage’s omelet is solved via the regret theory

Eq. (1) with $\\{p_{k}=q_{k}\\}_{k=1}^{n}$ can refer to the to the decision
model which assumes that at the moment of action-taking there is an uncertain
state of nature (environment) ${\cal S}_{k}$ to be realized from $\\{{\cal
S}_{k}\\}_{k=1}^{n}$ with probabilities $\\{p_{k}\\}_{k=1}^{n}$, which are
known to the decision maker luce ; jeffrey . ${\cal S}_{k}$ are called states
of nature, since their future realization is independent from the action
taken, but an action $A$ ($B$) in a state ${\cal S}_{k}$ leads to consequences
with monetary outcome $x_{k}$ ($y_{k}$) and utilities $u(x_{k})$ ($u(y_{k})$)
luce ; jeffrey ; cf. (1, 20).

The following classic decision problem is described in savage : A decision
maker has to finish making an omelet began by his wife, who has already broken
into a bowl five good eggs. A sixth unbroken egg is lying on the table, and it
must be either used in making the omelet, or discarded. There are two states
of the nature: good (the sixth egg is good) and rotten (the sixth egg is
rotten), which do not depend on the actions $A_{1}$, $A_{2}$ and $A_{3}$ of
the decision maker.

$A_{1}$: break the sixth egg into the bowl.

$A_{2}$: discard the sixth egg.

$A_{3}$: break the sixth egg into a saucer; add it to the five eggs if it is
good, discard it if it is rotten.

The consequences of the acts can be written as lotteries:

$\displaystyle A_{1}=\begin{pmatrix}u_{-5}&u_{6}\\\ p&1-p\end{pmatrix},\qquad
A_{2}=\begin{pmatrix}u_{5}&u_{5}+z\\\ p&1-p\end{pmatrix},\qquad
A_{3}=\begin{pmatrix}u_{5}+w&u_{6}+w\\\ p&1-p\end{pmatrix},$ (46)

where $p$ ($1-p$) is the objective probability for the sixth egg to be rotten
(good), $u_{6}$ ($u_{5}$) is the utility of the six-egg (five-egg) omelet,
$u_{-5}<0$ is the utility of five spoiled eggs and no omelet whatsoever, $w<0$
is the utility of washing the saucer, and $z<0$ is the utility of the good egg
being lost. 555The concrete utilities of washing the saucer may differ
depending on the state of the sixth egg. We, however, neglect this difference.
Also, for simplicity $w$ was simply added to $u_{5}$ and $u_{6}$.

Now looking at the consequences of $A_{2}$, we see that—in contrast to $A_{1}$
and $A_{3}$—acting $A_{2}$ does not resolve the uncertain state of nature:
once the egg is discarded, the decision maker will not know (without
additional actions), whether it was rotten or good. Put differently, utility
$z$ is not obtained after acting $A_{2}$, and cannot be obtained without
additional actions. Calculating the expecting utility of $A_{2}$ in the usual
way as $pu_{5}+(1-p)(u_{5}+z)$ does not apply, because it disregards this
aspect $A_{2}$. It is natural to take the expected utility as
$pu_{5}+(1-p)u_{5}=u_{5}$ (i.e. once $z$ is not obtained, it is not included),
but then comparing with expected utilities of $A_{1}$ and $A_{3}$, we see that
the parameter $z$ will appear nowhere. Hence, we suggest that the expected
utility does not apply to comparing $A_{2}$ with the other two actions.

Employing in (46) the reasoning of regret [cf. (8, 18, 20)] does take into
account the difference between $A_{2}$ and the other two actions. Let us for
example calculate the regret about not taking $A_{1}$ once $A_{2}$ has been
taken:

$\displaystyle R(A_{1},A_{2})=pf(u_{-5}-u_{5})+(1-p)f(u_{6}-u_{5}).$ (47)

This expression does not contain $z$, since the uncertain state of nature was
not resolved after acting $A_{2}$, i.e. after acting $A_{2}$ the obtained
utility is $u_{5}$.

On the other hand, acting $A_{1}$ resolves the uncertainty about the state of
nature. Hence the regret of not taking $A_{2}$, once $A_{1}$ was acted reads:

$\displaystyle R(A_{2},A_{1})=pf(u_{5}-u_{-5})+(1-p)f(u_{5}+z-u_{6}),$ (48)

i.e. once $A_{1}$ is taken and the egg is rotten (good), then the decision
maker already knows that if $A_{2}$ would be taken, then the egg will turn out
rotten (good). It is seen that (48) contains $z$ (the utility of discarding a
good egg), while (47) does not. Now

$\displaystyle A_{1}\succeq_{\rm reg}A_{2}\quad{\rm iff}\quad
R(A_{2},A_{1})\leq R(A_{1},A_{2}),$ (49)

where $R(A_{2},A_{1})-R(A_{1},A_{2})$ does feel the parameter $z$. As an
example of (49) consider $f(x)=x$ [cf. the discussion after (15)]:

$\displaystyle p(u_{5}-u_{-5})<(1-p)(u_{6}-u_{5}-\frac{z}{2}).$ (50)

where we recall that $u_{6}>u_{5}>u_{-5}$ and $z<0$. We can naturally assume
$u_{5}-u_{-5}>u_{6}-u_{5}>0$ under which (50) is non-trivial even for $p=1/2$.
Note that the formal application of the expected utility will claim that
$A_{1}$ is preferred over $A_{2}$ for $p(u_{5}-u_{-5})<(1-p)(u_{6}-u_{5}-z)$,
which is clearly different from (50). This is not just a different outcome;
rather, the expected utility does not apply.

##  VI Summary

This paper studied regret functionals over the utility differences of two
probabilistic lotteries; see section II. There are various types of lotteries,
from independent to fully dependent that refer to the same state of nature.
The regret functional compares the lotteries counter-factually taking notice
of their probabilities. For particular cases, the regret reverts to the
expected utility. More generally, it does not satisfy the independence (from
irrelevant alternatives) axiom of the expected utility, also known as the sure
thing principle. This not satisfying is by itself non-trivial and is explored
in Appendix B. In contrast to the expected utility, the regret is also
generally not invariant with respect to multiplying the utility by a positive
number. It is also due to these two differences compared to the expected
utility that the regret is efficient for explaining and resolving Allais’s
paradox; see section III. The resolution demands a non-trivial features of the
regret functional, viz. its super-additivity, which does make an intuitive
sense. We show that the regret functional can be chosen such that the regret-
ordering holds transitivity. In particular, Allais’s paradox can be resolved
via a transitive regret, and this resolution provides a consistent account of
changes in monetary outcomes.

We devoted a special attention to relations between (the first-order)
stochastic dominance and the regret-preference; see section IV. The former
ordering is normatively appealing, but it is incomplete, since not every two
lotteries can be compared with each other. We show that for independent
lotteries the stochastic dominance implies the regret-preference. For
dependent lotteries the relations between the two are more complex. Here we
proposed a sufficient condition for the implication stochastic dominance $\to$
regret-preference, which, interestingly is also based on the super-additivity
of the regret; see Proposition 5.

Finally, we show in section V how the considered regret theory can be useful
in those situations, where actions of the decision maker do not resolve the
uncertain situation. The expected utility theory does not apply to such a
situation in the sense that there is an important information about the
lotteries that it simply discards. In the regret, this information is
employed, since the regret compares the unresolved uncertainty with the
resolved uncertainty.

Our results show that though the concept of regret was initially deduced from
certain emotional features of decision makers, it does have many features one
intuitively expects from rationality. Hence we envisage its further
applications in e.g. reinforcement learning.

## Acknowledgements

This work was supported by State Science Committee of Armenia, grants No.
21AG-1C038. We thank Andranik Khachatryan for useful remarks and for
participating in initial stages of this work.

## References

  * (1) S. C. Stearns, “Daniel bernoulli (1738): evolution and economics under risk,” Journal of biosciences, vol. 25, no. 3, pp. 221–228, 2000.
  * (2) R. D. Luce and H. Raiffa, Games and decisions: Introduction and critical survey.  Courier Corporation, 1989.
  * (3) L. J. Savage, The foundations of statistics.  Courier Corporation, 1972.
  * (4) J. D. Hey, “Experiments and the economics of individual decision making under risk and uncertainty,” 1997.
  * (5) J. Baron, Thinking and deciding.  Cambridge University Press, 2000.
  * (6) J. Hofbauer, K. Sigmund, et al., Evolutionary games and population dynamics.  Cambridge university press, 1998.
  * (7) T. Everitt, J. Leike, and M. Hutter, “Sequential extensions of causal and evidential decision theory,” in International Conference on Algorithmic Decision Theory, pp. 205–221, Springer, 2015.
  * (8) M. Allais and G. Hagen, Expected utility hypotheses and the Allais paradox: Contemporary discussions of the decisions under uncertainty with Allais’ rejoinder, vol. 21.  Springer Science & Business Media, 2013.
  * (9) M. J. Machina, “Choice under uncertainty: Problems solved and unsolved,” in Foundations of Insurance economics, pp. 49–82, Springer, 1992.
  * (10) D. Kahneman and A. Tversky, “Prospect theory: An analysis of decision under risk,” in Handbook of the fundamentals of financial decision making: Part I, pp. 99–127, World Scientific, 2013.
  * (11) A. Tversky and D. Kahneman, “Advances in prospect theory: Cumulative representation of uncertainty,” Journal of Risk and uncertainty, vol. 5, no. 4, pp. 297–323, 1992.
  * (12) J. Quiggin, Generalized expected utility theory: The rank-dependent model.  Springer Science & Business Media, 2012.
  * (13) E. Diecidue and P. P. Wakker, “On the intuition of rank-dependent utility,” Journal of Risk and Uncertainty, vol. 23, no. 3, pp. 281–298, 2001.
  * (14) S. Bourgeois-Gironde, “Regret and the rationality of choices,” Philosophical Transactions of the Royal Society B: Biological Sciences, vol. 365, no. 1538, pp. 249–257, 2010.
  * (15) M. H. Acker, “Tempered regrets under total ignorance,” Theory and Decision, vol. 42, no. 3, pp. 207–213, 1997.
  * (16) G. Loomes and R. Sugden, “Regret theory: An alternative theory of rational choice under uncertainty,” The economic journal, vol. 92, no. 368, pp. 805–824, 1982.
  * (17) D. E. Bell, “Regret in decision making under uncertainty,” Operations research, vol. 30, no. 5, pp. 961–981, 1982.
  * (18) H. Bleichrodt and P. P. Wakker, “Regret theory: A bold alternative to the alternatives,” The Economic Journal, vol. 125, no. 583, pp. 493–532, 2015\.
  * (19) P. C. Fishburn, “An axiomatic characterization of skew-symmetric bilinear functionals, with applications to utility theory,” Economics Letters, vol. 8, no. 4, pp. 311–313, 1981.
  * (20) R. Sugden, “An axiomatic foundation for regret theory,” Journal of Economic Theory, vol. 60, no. 1, pp. 159–180, 1993.
  * (21) C. Starmer, “Developments in non-expected utility theory: The hunt for a descriptive theory of choice under risk,” Journal of economic literature, vol. 38, no. 2, pp. 332–382, 2000.
  * (22) J. Quiggin, “Stochastic dominance in regret theory,” The Review of Economic Studies, vol. 57, no. 3, pp. 503–511, 1990.
  * (23) H. Levy, “Regret theory: State dominance and expected utility,” Journal of Mathematical Psychology, vol. 79, pp. 1–12, 2017.
  * (24) R. C. Jeffrey, The logic of decision.  University of Chicago press, 1990.
  * (25) J. Pearl, “The sure-thing principle,” Journal of Causal Inference, vol. 4, no. 1, pp. 81–86, 2016.
  * (26) S. Bikhchandani and U. Segal, “Transitive regret,” Theoretical Economics, vol. 6, no. 1, pp. 95–108, 2011.
  * (27) P. C. Fishburn, “Nontransitive preferences in decision theory,” Journal of risk and uncertainty, vol. 4, no. 2, pp. 113–134, 1991.
  * (28) S. Bikhchandani and U. Segal, “Transitive regret over statistically independent lotteries,” Journal of Economic Theory, vol. 152, pp. 237–248, 2014.
  * (29) P. C. Fishburn, “Transitive measurable utility,” Journal of Economic Theory, vol. 31, no. 2, pp. 293–317, 1983.
  * (30) P. J. Schoemaker, “The expected utility model: Its variants, purposes, evidence and limitations,” Journal of economic literature, pp. 529–563, 1982.
  * (31) Y. Tanaka and M. Hattori, “Convexity, concavity, super-additivity, and sub-additivity of cost function without fixed cost,” 2017.
  * (32) B. J. Weber et al., “The effects of payout and probability magnitude on the allais paradox,” Memory & cognition, vol. 36, no. 5, pp. 1013–1023, 2008.
  * (33) C. F. Camerer, “An experimental test of several generalized utility theories,” Journal of Risk and uncertainty, vol. 2, no. 1, pp. 61–104, 1989.
  * (34) P. R. Blavatskyy, A. Ortmann, and V. Panchenko, “Now you see it, now you don’t: How to make the allais paradox appear, disappear, or reverse,” UNSW Business School Research Paper, no. 2015-14, 2015.
  * (35) E. Incekara-Hafalir, E. Kim, and J. D. Stecher, “Is the allais paradox due to appeal of certainty or aversion to zero?,” Experimental Economics, vol. 24, no. 3, pp. 751–771, 2021.

##  Appendix A Proof of Proposition 4

.

We are to show that

$\sum_{i=1,j=1}^{n}p_{i}q_{j}g(u(x_{j})-u(x_{i}))\leq 0.$ (51)

It can be written in the following form

$\sum_{i=1}^{n}\sum_{j=1}^{i}(p_{i}q_{j}-p_{j}q_{i})g(u(x_{j})-u(x_{i})).$
(52)

Now $g$’s in (52) are strictly negative (recall (40) and monotonicity of
$g(x)$ and $u(x)$).

We show that the following holds

$\begin{split}\sum_{i=1}^{n}\sum_{j=1}^{i}p_{i}q_{j}g(u(x_{j})-u(x_{i}))&\leq\sum_{i=1}^{n}\sum_{j=1}^{i}p_{i}p_{j}g(u(x_{j})-u(x_{i}))\\\
&\leq\sum_{i=1}^{n}\sum_{j=1}^{i}p_{j}q_{i}g(u(x_{j})-u(x_{i})),\end{split}$
(53)

from which the (52) will follow.

To do it, we make use of the following lemma.

Lemma. Given sequences $p_{i}$, $q_{i}$ as above, then for any increasing
sequence of negative numbers $g_{i}$, it is true that

$\sum_{i=1}^{k}p_{i}g_{i}\geq\sum_{i=1}^{k}q_{i}g_{i},\quad{\rm for}\quad
k=1,...,n.$ (54)

A simple induction will help. Obviously $p_{1}g_{1}\geq q_{1}g_{1}$. Supposing
it is true for some $k$ we will have:

$\sum_{i=1}^{k}p_{i}g_{i}\geq\sum_{i=1}^{k}q_{i}g_{i}$ (55)

Let us now subtract $g_{k+1}$ from each component $g_{i}$. (As we are speaking
about any increasing sequence, we haven’t specified any concrete $g$-s yet.
So, having that all components are less $g_{k+1}$, subtract, without change of
sign in inequality)

We will have the following:

$\sum_{i=1}^{k}p_{i}(g_{i}-g_{k+1})\geq\sum_{i=1}^{k}q_{i}(g_{i}-g_{k+1})$
(56)

Now by (41)

$\sum_{i=1}^{k+1}p_{i}\leq\sum_{i=1}^{k+1}q_{i}$ (57)

multiplying by $g_{k+1}$, we get

$\sum_{i=1}^{k+1}p_{i}g_{k+1}\geq\sum_{i=1}^{k+1}q_{i}g_{k+1}.$ (58)

Summing up (56) and (58), the desired result is obtained:

$\sum_{i=1}^{k+1}p_{i}g_{i}\geq\sum_{i=1}^{k+1}q_{i}g_{i}.$ (59)

Using the lemma we go back and consider first part of (53)

$\sum_{i=1}^{n}\sum_{j=1}^{i}p_{i}q_{j}g(u(x_{j})-u(x_{i}))\leq\sum_{i=1}^{n}\sum_{j=1}^{i}p_{i}p_{j}g(u(x_{j})-u(x_{i})).$
(60)

Obviously

$\begin{split}\sum_{i=1}^{n}\sum_{j=1}^{i}p_{i}q_{j}g(u(x_{j})-u(x_{i}))&=\sum_{i=1}^{n}p_{i}\sum_{j=1}^{i}q_{j}g(u(x_{j})-u(x_{i}))\\\
&\leq\sum_{i=1}^{n}p_{i}\sum_{j=1}^{i}p_{j}g(u(x_{j})-u(x_{i})),\end{split}$
(61)

The last part is implied by lemma.

Returning to the second part of (53):

$\sum_{i=1}^{n}\sum_{j=1}^{i}p_{j}q_{i}g(u(x_{j})-u(x_{i}))\geq\sum_{i=1}^{n}\sum_{j=1}^{i}p_{i}p_{j}g(u(x_{j})-u(x_{i})).$
(62)

and changing the order of summation, we get

$\begin{split}\sum_{i=1}^{n}&\sum_{j=1}^{i}p_{j}q_{i}g(u(x_{j})-u(x_{i}))=\sum_{j=1}^{n}\sum_{i=j}^{n}p_{j}q_{i}g(u(x_{j})-u(x_{i}))\\\
&=\sum_{j=1}^{n}p_{j}\sum_{i=j}^{n}q_{i}g(u(x_{j})-u(x_{i}))\geq\sum_{j=1}^{n}p_{j}\sum_{i=j}^{n}p_{i}g(u(x_{j})-u(x_{i}))\\\
&=\sum_{i=1}^{n}\sum_{j=1}^{i}p_{i}p_{j}g(u(x_{j})-u(x_{i})),\end{split}$ (63)

where the inequality is the inverse of the one used in lemma. The proof is
complete.

##  Appendix B Regret and the independence axiom (7)

We already emphasized around (38, 39) that regret preference $\succeq_{\rm
reg}$ defined in (11) must violate the independence axiom for solving Allais’
paradox. Now we provide clarifications regarding the form (7) of this axiom.
We note that (6, 7) do not define how precisely the mixing of the two
probabilities with weights $\alpha$ is implemented. Below we discuss three
interesting possibilities for implementing the set-up of (7). So we are given
three lotteries [cf. (1)]

$\displaystyle(x,p)=\begin{pmatrix}x_{1}&x_{2}&...&x_{n}\\\
p_{1}&p_{2}&...&p_{n}\end{pmatrix},\qquad(y,q)=\begin{pmatrix}y_{1}&y_{2}&...&y_{n}\\\
q_{1}&q_{2}&...&q_{n}\end{pmatrix},\qquad(z,r)=\begin{pmatrix}z_{1}&z_{2}&...&z_{n}\\\
r_{1}&r_{2}&...&r_{n}\end{pmatrix},$ (64)
$\displaystyle{\sum}_{k=1}^{n}p_{k}={\sum}_{k=1}^{n}q_{k}={\sum}_{k=1}^{n}r_{k}=1,$
(65)

where $(p_{1},...,p_{n})$, $(q_{1},...,q_{n})$, and $(r_{1},...,r_{n})$ are
(resp.) the probabilities of monetary outcomes $(x_{1},...,x_{n})$ and
$(y_{1},...,y_{n})$, $(z_{1},...,z_{n})$ within each lottery.

1. Here one chooses between two composite lotteries $A=\\{(1-\alpha)(x,p)+\alpha(z,r)\\}$ and $B=\\{(1-\alpha)(y,q)+\alpha(z,r)\\}$. If $A$ is taken, then a binary random variable $S_{A}$ is realized that takes values $S_{A}=0$ and $S_{A}=1$ with probabilities $1-\alpha$ and $\alpha$, respectively. For $S_{A}=0$ or $S_{A}=1$ one faces lottery $(x,p)$ or $(z,r)$, respectively. If $B$ is taken, then a binary random variable $S_{B}$ (independent from $S_{A}$) is realized that takes values $S_{B}=0$ and $S_{B}=1$ with probabilities $1-\alpha$ and $\alpha$, respectively. For $S_{B}=0$ or $S_{B}=1$ one faces lottery $(y,q)$ or $(z,r)$, respectively. Let us now assume that $(x,p)$ is independent from $(y,q)$, but the lottery $(z,r)$ in both options is the same. Using definition of regret (11), we end up with the following preference relation:

$\displaystyle A\succeq_{\rm reg,1}B\quad{\rm
iff}\quad(1-\alpha)^{2}{\sum}_{i,j=1}^{n}g(u(y_{j})-u(x_{i}))p_{i}q_{j}+(1-\alpha)\alpha{\sum}_{i,j=1}^{n}g(u(z_{j})-u(x_{i}))p_{i}r_{j}$
$\displaystyle+(1-\alpha)\alpha{\sum}_{i,j=1}^{n}g(u(y_{j})-u(z_{i}))q_{j}r_{i}\leq
0.$ (66)

Note that (66) does not contain terms with $g(u(z_{j})-u(z_{i}))$, because the
decision maker does not expect to find different outcomes $z_{j}$ and $z_{i}$
within options $A$ and $B$.

2. Now we have the situation of 1, but $S=S_{A}=S_{B}$; e.g. one can assume that $S$ is realized beforehand, but the result is not known to the decision maker at the time of decision making. Now the regret is different [cf. (66)]:

$\displaystyle A\succeq_{\rm reg,2}B\quad{\rm
iff}\quad(1-\alpha)^{2}{\sum}_{i,j=1}^{n}g(u(y_{j})-u(x_{i}))p_{i}q_{j}\leq
0,$ (67)

where $(z,r)$ does not enter to regret comparison (67), which is formally
consistent with axiom (7).

3. We have the situation of 1, but $(z,r)$ in option $A$ and $(z,r)$ in option $B$ are two different lotteries with independent probabilities:

$\displaystyle A\succeq_{\rm reg,3}B\quad{\rm
iff}\quad(1-\alpha)^{2}{\sum}_{i,j=1}^{n}g(u(y_{j})-u(x_{i}))p_{i}q_{j}+(1-\alpha)\alpha{\sum}_{i,j=1}^{n}g(u(z_{j})-u(x_{i}))p_{i}r_{j}$
$\displaystyle+(1-\alpha)\alpha{\sum}_{i,j=1}^{n}g(u(y_{j})-u(z_{i}))q_{j}r_{i}+\alpha^{2}{\sum}_{i,j=1}^{n}g(u(z_{j})-u(z_{i}))r_{j}r_{i}\leq
0.$ (68)

The standard interpretation of the independence axiom within the expected
utility theory hints at 3. We however emphasized that this situation is not
unique. Note that all possibilities (66, 67, 68) agree with the expected
utility theory, where $g(x)=x$.

##  Appendix C Proof of Proposition 5

It is known that $n\times n$ matrices $\theta_{i,j}$ from (43) form vector
space of $(n-1)\times(n-1)$ dimension. So for example any such $3\times 3$
matrix can be rewritten

$\begin{pmatrix}\theta_{1,1}&\theta_{1,2}&\theta_{1,3}\\\
\theta_{2,1}&\theta_{2,2}&\theta_{2,3}\\\
\theta_{3,1}&\theta_{3,2}&\theta_{3,3}\end{pmatrix}=\theta_{1,1}\begin{pmatrix}1&0&-1\\\
0&0&0\\\ -1&0&1\end{pmatrix}+\theta_{1,2}\begin{pmatrix}0&1&-1\\\ 0&0&0\\\
0&-1&1\end{pmatrix}+\theta_{2,1}\begin{pmatrix}0&0&0\\\ 1&0&-1\\\
-1&0&1\end{pmatrix}+\theta_{2,2}\begin{pmatrix}0&0&0\\\ 0&1&-1\\\
0&1&1\end{pmatrix}$ (69)

Denoting the basis matrices by $M_{i,j}$ we have that any matrix $\Theta$ of
thetas can be rewritten as

$\displaystyle\Theta=\sum_{i,j=1}^{n-1}\theta_{i,j}M_{i,j},$ (70)

where $M_{i,j}$ is the matrix whose $(i,j)$-th and $(n,n)$-th elements are
$1$, the $(i,n)$-th and $(n,j)$-th elements are $-1$.

Let us compute the regret in this case

$R=\sum_{i=1}^{n}\sum_{j=1}^{n}P(x_{i},x_{j})g(u(x_{j})-u(x_{i}))=\sum_{i=1}^{n}\sum_{j=1}^{n}p_{i}q_{j}g(u(x_{j})-u(x_{i}))+\sum_{i=1}^{n}\sum_{j=1}^{n}\theta_{i,j}g(u(x_{j})-u(x_{i}))$
(71)

We already know that the first term is negative ((51) and Proposition 4). So
it remains to show that second part is also negative. Denoting by $G$ the
matrix whose elements are $G(i,j)=g(u(x_{j})-u(x_{i}))$, we can rewrite

$\sum_{i=1}^{n}\sum_{j=1}^{n}\theta_{i,j}g(u(x_{j})-u(x_{i}))=||\Theta\odot
G||,$ (72)

Where under $||.||$ we understand sum of all elements and $\odot$ is
Hadamard’s (element-wise) product.

Note that

$\sum_{i=1}^{n}\sum_{j=1}^{n}\theta_{i,j}g(u(x_{j})-u(x_{i}))=||\Theta\odot
G||=\sum_{i=1}^{n-1}\sum_{j=1}^{n-1}\theta_{i,j}||M_{i,j}\odot G||.$ (73)

We have

$||M_{i,j}\odot
G||=g(u(x_{i})-u(x_{j}))-g(u(x_{n})-u(x_{j}))-g(u(x_{i})-u(x_{n})).$ (74)

Now consider

$\begin{split}&\theta_{i,j}||M_{i,j}\odot G||+\theta_{j,i}||M_{i,j}\odot
G||\\\
&=\theta_{i,j}(g(u(x_{i})-u(x_{j}))-g(u(x_{n})-u(x_{j}))-g(u(x_{i})-u(x_{n})))\\\
&+\theta_{j,i}(g(u(x_{j})-u(x_{i}))-g(u(x_{n})-u(x_{i}))-g(u(x_{j})-u(x_{n})))\\\
&=(\theta_{i,j}-\theta_{j,i})(g(u(x_{i})-u(x_{j}))+g(u(x_{n})-u(x_{i}))-g(u(x_{n})-u(x_{j}))\end{split}$
(75)

Note that while $i>j$ we have by super-additivity that term in second
parenthesis is negative. So toghether with $\theta_{i,j}\geq\theta_{j,i}$ we
conclude that

$\theta_{i,j}||M_{i,j}\odot G||+\theta_{j,i}||M_{i,j}\odot G||\leq 0$ (76)

Rewriting

$\sum_{i=1}^{n}\sum_{j=1}^{n}\theta_{i,j}g(u(x_{j})-u(x_{i}))=||\Theta\odot
G||=\sum_{i=1}^{n-1}\sum_{j=1;j\neq i}^{n-1}\theta_{i,j}||M_{i,j}\odot
G||+\sum_{i=1}^{n-1}\theta_{i,i}||M_{i,i}\odot G||.$ (77)

The second sum in (77) is obviously 0, as $g(x)$ is antisymmetric, and
$M_{i,i}$-s are symmetric matrices.

So

$\sum_{i=1}^{n}\sum_{j=1}^{n}\theta_{i,j}g(u(x_{j})-u(x_{i}))=\sum_{i=1}^{n-1}\sum_{j=1;j\neq
i}^{n-1}\theta_{i,j}||M_{i,j}\odot G||\leq 0.$ (78)