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\begin{document}
\title{Game Theory without Decision-Theoretic Paradoxes}
\author{Pierfrancesco La Mura}
\institute{Department of Economics and Information Systems\\
HHL - Leipzig Graduate School of Management\\
Jahnallee 59, 04109 Leipzig (Germany)\\
plamura@hhl.de\\
}
\maketitle

\begin{abstract}
%\begin{quote}
Most work in game theory is conducted under the assumption that the players are expected utility maximizers. Expected utility is a very tractable decision model, but is prone to well-known paradoxes and empirical violations (Allais 1953, Ellsberg 1961), which may induce systematic biases in game-theoretic predictions. La Mura (2009) introduced a projective generalization of expected utility (PEU) which avoids the dominant paradoxes, while remaining quite tractable. We show that every finite game with PEU players has an equilibrium, and discuss several examples of PEU games.
\end{abstract} 
Keywords: game theory, expected utility, paradoxes, Allais, Ellsberg, projective

\section{Introduction} The expected utility hypothesis is the
\emph{de facto} foundation of game theory. The von Neumann - Morgenstern axiomatization of expected utility, and later on
the subjective formulations by Savage (1954) and Anscombe and Aumann (1963)
were immediately greeted as simple and intuitively compelling. Yet, in the
course of time, a number of empirical violations and paradoxes (Allais 1953,
Ellsberg 1961) came to cast doubt on the validity of
the hypothesis as a foundation for the theory of rational decisions in
conditions of risk and subjective uncertainty. In economics and in the social
sciences, the shortcomings of the expected utility hypothesis are generally
well-known, but often tacitly accepted in view of the great tractability and
usefulness of the corresponding mathematical framework. In fact, the
hypothesis postulates that preferences can be represented by way of a utility
functional which is linear in probabilities, and linearity makes expected
utility representations particularly tractable in models and applications. 
In particular, in a game-theoretic context, linearity of expected utility ensures 
that the best response correspondence is also linear, and hence that any finite game has a 
Nash equilibrium. 

In economics and the social sciences, the importance of accounting for violations of the
expected utility hypothesis has long been recognized (Tversky 1975), but so 
far none of its numerous alternatives (e.g., Machina 1982, Schmeidler 1989, to
quote only two particularly influential papers in a rich and constantly evolving 
literature) has fully succeeded in replacing expected utility as a standard foundation 
for decisions under uncertainty, partly due to the great mathematical tractability 
of expected utility relative to many of its proposed generalizations. 

La Mura (2009) introduced projective expected utility (PEU), a decision-theoretic framework which 
accommodates the dominant paradoxes while retaining significant simplicity and tractability. 
This is obtained by weakening the expected utility 
hypothesis to its projective counterpart, in analogy with the quantum-mechanical 
generalization of classical probability theory. We first review the EU and PEU frameworks, and show that the latter is sufficiently general to avoid 
both Allais' and Ellsberg's paradoxes. We then extend the notion of Nash equilibrium to games with PEU preferences,
and prove that any finite game with PEU players has an equilibrium. Finally, we discuss several examples of games with
PEU preferences and identify observable deviations from the theory of games with EU preferences.

\section{von Neumann - Morgenstern Expected Utility} Let $\Omega$ be a finite set
of outcomes, and $\Delta$ be the set of probability functions defined on $\Omega,$
taken to represent risky prospects (or {\emph{lotteries}}). Next, let
$\succsim$ be a complete and transitive binary relation defined on $\Delta
\times \Delta,$ representing a decision-maker's preference ordering over
lotteries. Indifference of $p, q \in \Delta$ is defined as [$p \succsim q$ and
$q \succsim p$] and denoted as $p \sim q$, while strict preference of $p$ over
$q$ is defined as [$p \succsim q$ and not $q \succsim p$], and denoted by $p
\succ q$. The preference ordering is assumed to satisfy the following two
conditions.

\begin{axiom}
  (Archimedean) For all $p, q, r \in \Delta$ with $p \succ q \succ r,$ there
  exist $\alpha, \beta \in (0, 1)$ such that $\alpha p + (1 - \alpha) r \succ
  q \succ \beta p + (1 - \beta) r.$
\end{axiom}

\begin{axiom}
  (Independence) For all $p, q, r \in \Delta,$ $p \succsim q$ if, and only if,
  $\alpha p + (1 - \alpha) r \succsim \alpha q + (1 - \alpha) r$ for all
  $\alpha \in [0, 1] .$
\end{axiom}

{\smallskip}

A functional $u : \Delta \rightarrow \mathbb{R}$ is said to represent $\succsim$ if,
for all $p, q \in \Delta,$ $p \succsim q$ if and only if $u (p) \geq u (q) .$

\begin{theorem}
  (von Neumann and Morgenstern) Axioms $1$ and $2$ are jointly equivalent to
  the existence of a functional $u : \Delta \rightarrow \mathbb{R}$ which represents
  $\succsim$ such that, for all $p \in \Delta$,
  \[ u (p) = \sum_{\omega \in \Omega} u (\omega) p (\omega) . \]
\end{theorem}

The von Neumann - Morgenstern setting is appropriate whenever the nature of
the uncertainty is purely objective: all lotteries are associated with objective
random devices, such as dice or roulette wheels, with well-defined and known
frequencies for all outcomes, and the decision-maker only evaluates a lottery
based on the frequencies of its outcomes.

\section{Allais' Paradox} The following paradox
is due to Allais (1953). First, please choose between

\smallskip

A: A chance of winning $4000$ dollars with probability $0.2$

B: A chance of winning $3000$ dollars with probability $0.25$.

{\smallskip}

Now suppose that, instead of A and B, your two alternatives are

{\smallskip}

C: A chance of winning $4000$ dollars with probability $0.8$

D: A chance of winning $3000$ dollars with certainty.

{\smallskip}

If you chose $A$ over $B,$ and $D$ over $C,$ then you are in the modal class
of respondents. The paradox lies in the observation that $A$ and $C$ are
special cases of a two-stage lottery $E$ which in the first stage either
returns zero dollars with probably $(1 - \alpha)$ or, with probability
$\alpha,$ leads to a second stage where one gets $4000$ dollars with
probability $0.8$ and zero otherwise. In particular, if $\alpha$ is set to $1$
then $E$ reduces to $C,$ and if $\alpha$ is set to $0.25$ it reduces to $A.$
Similarly, $B$ and $D$ are special cases of a two-stage lottery $F$ which
again with probability $(1 - \alpha)$ returns zero, and with probability
$\alpha$ continues to a second stage where one wins $3000$ dollars with
probability $1.$ Again, if $\alpha = 1$ then $F$ reduces to $D,$ and if
$\alpha = 0.25$ it reduces to $B.$ Then it is easy to see that the $[A \succ
B,$ $D \succ C]$ pattern violates Axiom 2 (Independence), as $E$ can be
regarded as a lottery $\alpha p + (1 - \alpha) r$, and $F$ as a lottery
$\alpha q + (1 - \alpha) r$, where $p$ and $q$ represent lottery $C$ and $D$,
respectively, and $r$ represents the lottery in which one gets zero dollars
with certainty. When comparing $E$ and $F$, why should it matter what is the
value of $\alpha ?$ Yet, experimentally one finds that it does.

\section{Ellsberg's Paradox} Another disturbing violation of the expected
utility hypothesis was pointed out by Ellsberg (1961). Suppose that an urn
contains $300$ balls of three possible colors: red, green, and
blue. You know that the urn contains exactly $100$ red balls, but are
given no information on the proportion of green and blue.

You win if you guess which color will be drawn. Do you prefer to bet on red
($R$) or on green ($G$)? Many respondents choose $R$, on grounds that the
probability of drawing a red ball is known to be $1 / 3$, while the only
information on the probability of drawing a green ball is that it is between
$0$ and $2 / 3.$ Now suppose that you win if you guess which color will
\emph{not} be drawn. Do you prefer to bet that red will not be drawn
($\overline{R}$) or that green will not be drawn ($\overline{G}$)? Again many
respondents prefer to bet on $\overline{R}$, as the probability is known ($2 /
3$) while the probability of $\overline{G}$ is only known to be between $1$
and $1 / 3.$

The pattern [$R \succ G$, $\overline{R} \succ \overline{G}$] is incompatible
with von Neumann - Morgenstern expected utility, which only deals with known
probabilities, and is also incompatible with the Savage (1954) formulation of
expected utility with subjective probability as it violates its Sure Thing
axiom. 
Observe that, in Ellsberg's setting, the decision-maker ignores the actual composition 
of the urn, and hence operates in a state of subjective uncertainty about the 
true probabilistic state of affairs. In particular, the decision-maker is 
exposed to a combination of subjective uncertainty (on the actual composition of the urn) 
and objective risk (the probability of drawing a specific color from an urn 
of given composition). The paradox suggests that, in order to account for 
the choice pattern discussed above, subjective uncertainty and 
risk should be  handled as distinct notions.

\section{Projective Expected Utility} 

Let $X$ be the positive orthant of the
unit sphere in $\mathbb{R}^n$, where $n$ is the cardinality of the set of
relevant outcomes $\Omega.$ Then von
Neumann - Morgenstern lotteries, regarded as elements of the unit simplex, are
in one-to-one correspondence with elements of $X,$ which can therefore be
interpreted as risky prospects, for which the frequencies of the relevant
outcomes are fully known. Observe that, while the projections of elements of
the unit simplex (and hence, $L^1$ unit vectors) on the basis vectors can be
naturally associated with probabilities, if we choose to model von Neumann -
Morgenstern lotteries as elements of the unit sphere (and hence, as unit
vectors in $L^2$) then probabilities are naturally associated with
\emph{squared} projections. The advantage of such move is that $L^2$ is
the only $L^p$ space which is also a Hilbert space, and Hilbert spaces have a
very tractable projective structure which is exploited by the representation.
In particular, it is unique to $L^2$ that the set of unit vectors is invariant
with respect to projections.

Next, let $\left\langle .|. \right\rangle$ denote the usual inner product in
$\mathbb{R}^n$. We denote the transpose of a vector or a matrix with a prime, \textit{e.g.}, $x'$ denotes the transpose of $x$. An 
orthonormal basis is a set of unit vectors $\{ b^1, ...,
b^n \}$ such that $\left\langle b^i |b^j \right\rangle = 0$ whenever $i \neq j$.
In our context, orthogonality captures the idea that two events or outcomes
are mutually exclusive (for one event to have probability one, the other must
have probability zero). 
The natural basis corresponds to the set of degenerate lotteries returning
each objective lottery outcome with certainty, and is conveniently identified
with the set of objective lottery outcomes $\left\{ \omega^1, \omega^2, ..., \omega^n \right\}$. Yet, in any realistic experimental
setting, it is very unlikely that those objective outcomes will happen to
coincide with the set of subjective consequences which are relevant from the perspective of the
decision-maker. Moreover, even if the latter could be fully elicited, it would be
generally problematic to relate a von Neumann - Morgenstern lottery, which only
specifies the probabilities of the objective outcomes, with the probabilities 
induced on the subjective consequences, that is, the relevant 
dimensions of risk from the point of view of the decision-maker. The
perspective of the observer or modeler is inexorably bound to objectively
measurable entities, such as frequencies and prizes; by contrast, the
decision-maker thinks and acts based on subjective preferences and subjective
consequences, which in a revealed-preference context should be presumed to exist
while at the same time assumed, as a methodological principle, to be
unaccessible to direct measurement.

In this section we shall relax the assumption, implicit in the von Neumann - Morgenstern 
setting, that the objective outcomes and subjective consequences coincide, and replace it with the
weaker requirement that there exists a set of mutually exclusive, jointly exhaustive subjective consequences with 
respect to which the decision-maker evaluates each uncertain prospect. As we shall see such
weaker condition, together with the usual assumptions of completeness and transitivity of preferences, and the
Archimedean and Independence conditions from the von Neumann-Morgenstern treatment, jointly 
characterize representability in terms of a projective generalization of the expected utility functional.

Let $B:= \{ b^1, ..., b^n \}$ represent a set of $n$ mutually exclusive, jointly exhaustive subjective consequences, identified with an orthonormal basis in $\mathbb{R}^n$. For any lottery $x \in X$, its associated \textit{risk profile} $p_x$ with respect to $B$ is defined by
  \[ p_x (b^i) = \left\langle x|b^i \right\rangle^2, i = 1, \ldots, n \]
The risk profile of a lottery $x$ returns the probabilities induced on the subjective consequences by playing lottery $x$. Observe that the risk profile with respect to the natural basis simply returns the probabilities of the lottery outcomes. Hence, in the present setting a lottery is identified both in terms of objective outcomes and subjective consequences. Since the position of the subjective basis relative to the natural basis can be arbitrary, a lottery can exhibit any combination of risk profiles on outcomes and consequences. 

\begin{axiom} There exists an orthonormal basis $Z:= \{ z^1, ..., z^n \}$ such that any two lotteries 
  $x,y \in X$ are indifferent whenever their risk profiles with respect to $Z$ coincide.
\end{axiom}

Axiom 3 requires that there exists a set of $n$ mutually exclusive, jointly exhaustive subjective consequences such that any two lotteries are only evaluated based on their risk profiles, \textit{i.e.}, on the probabilities they induce on those subjective consequences. 
In the von Neumann - Morgenstern treatment, Axiom 3 is tacitly assumed to hold
with respect to the natural basis in $\mathbb{R}^n .$ This implicit assumption
amounts to the requirement that lotteries are only evaluated based on the
probabilities they induce on the objective lottery outcomes.

While the probabilities with respect to the natural basis 
represent the relevant dimensions of risk as perceived by the 
modeler or an external observer (that is, the risk associated with the occurrence
of the objective outcomes), the \emph{preferred basis} postulated in Axiom 3 is allowed
to vary across different decision-makers, capturing the idea that the subjectively relevant
dimensions of risk (that is, those pertaining to the actual subjective
consequences) may be perceived differently by different subjects.
One case in which the relevant dimensions of risk may differ across subjects 
is in the presence of portfolio effects. Such effects are difficult to exclude or control for in experimental settings, as the subject's portfolio is typically unaccessible to direct measurement. 

Axiom 3 presumes that subjective consequences and 
objective outcomes have the same cardinality; we relax this assumption
later on, in the subjective formulation.

Once an orthonormal basis Z is given, each lottery $x$ can be associated with a
 function $p_x : Z \rightarrow [0, 1],$ such that $p_x (z^i) =
\left\langle x|z^i \right\rangle^2$ for all $z^i \in Z.$ Let $B$ be the set of
all such risk profiles $p_x,$ for $x \in X,$ and let $\succsim$ be
the complete and transitive preference ordering induced on $B \times B$ by
preferences on the underlying lotteries. 

Note that a convex combination
$\alpha p_x + (1 - \alpha) p_y$, where $p_x$ and $p_y$ are risk profiles, is still a well-defined risk profile.
We interpret this
type of mixing as objective, while
subjective mixing will be later on captured by subjective probability over the 
underlying (Anscombe-Aumann) states. We postulate the following two
axioms, which mirror those in the von Neumann - Morgenstern treatment.

\begin{axiom}
  (Archimedean) For all $x, y, t \in X$ with $p (x) \succ p (y) \succ p (t),$
  there exist $\alpha, \beta \in (0, 1)$ such that $\alpha p (x) + (1 -
  \alpha) p (t) \succ p (y) \succ$ $\beta p (x) + (1 - \beta) p (t) .$
\end{axiom}

{\smallskip}

\begin{axiom}
  (Independence) For all $x, y, t \in X$, $p_x \succsim p_y$ if, and only if,
  $\alpha p_x + (1 - \alpha) p_t \succsim ap_y + (1 - \alpha) p_t$ for all
  $\alpha \in [0, 1] .$
\end{axiom}

Some observations are in order at this point. First, note that the two axioms 
above impose conditions solely on risk profiles, and not
on the underlying lotteries. This seems appropriate, as the decision-maker 
is not ultimately concerned with the risk associated to the objective outcomes, 
but only with the risk induced on the relevant subjective consequences. 
It is also worth noting that, while risk profiles are now defined with respect 
to subjective consequences, rather than objective outcomes as in the von Neumann - Morgenstern treatment, in our setting they are still interpreted as objective probability functions.

\begin{theorem} (La Mura 2009)
  Axioms 3-5 are jointly equivalent to the existence of a symmetric matrix $U$
  such that $u (x) : = x' Ux$ for all $x \in X$ represents $\succsim$.
\end{theorem}

\section{Subjective Formulation} 

The following formulation extends the representation to situations of subjective
uncertainty. First, we introduce the following setup and
notation.

$\mathcal{S}$ is a finite set of states of Nature.

$\left. \langle .|. \right\rangle$ denotes the usual inner product in
Euclidean space.

$\Omega$ is the natural basis in $\mathbb{R}^n$, identified with a finite set
$\{\omega^1, \ldots ., \omega^n \}$ of lottery outcomes (prizes).

$Z$ is an orthonormal basis in $\mathbb{R}^m$, with $m \geq n,$ identified
with a finite set of subjective consequences $\{z^1, \ldots ., z^m
\}$. $V$ is an arbitrary $(m \times n)$ matrix chosen so that, for all
$\omega^i$ in $\Omega,$ $V \omega^i $ is a unit vector in $\mathbb{R}^m$. Observe that 
$V$ is always well defined as long as $m \geq n$. When $m=n$, we conventionally set $V \equiv I$, where
$I$ is the $n \times n$ identity matrix.

Lotteries correspond to $L^2$ unit vectors $x \in \mathbb{R}^n_+$; $X$ is the
set of all lotteries.

Since $\Omega$ is the natural basis, 
$\left. \left. \right\langle \omega^i |x \right\rangle^2 = x^2_i$; 
this quantity is interpreted as $p (\omega^i |x)$.


The quantity $\left\langle z^j |Vx \right\rangle^2$ is
interpreted as $p (z^j | x)$, the conditional probability of subjective
consequence $z^j$ given lottery $x$. In particular, 
$\left\langle z^j |V \omega^i_{^{}} \right\rangle^2$ is
interpreted as $p (z^j | \omega^i)$, the conditional probability of subjective
consequence $z^j$ given the degenerate lottery which returns objective 
outcome $\omega^i_{^{}}$ for sure. 

Once the subjective consequences $z^j$ are specified, for any lottery $x$ one
can readily compute $p (\omega^i |x) = x_i^2$ and $p (z^j |x) = \left\langle z^j |Vx \right\rangle^2$. 
Moreover, given the latter probabilistic
constraints, one can readily identify a lottery $x$ and an
orthonormal basis $Z$ which jointly satisfy them. Hence, in the above
construction lotteries are identified with respect to two different frames of
reference: objective lottery outcomes, and subjective consequences.

Observe that $p (z^j | x)$ generally differs from the probability of $z^j$ given $x$ 
computed according to the law of total probability, which is given by 
$\sum_i p (\omega^i |x) p (z^j | \omega^i) = \sum_i x_i^2 \left\langle z^j |V
\omega^i_{^{}} \right\rangle^2$. To get a sense of why and how the law of total probability
may fail, let us consider a decision-maker who really hates to lose whenever 
the probability of winning is high (more so than when the probability of winning is low), 
and loves to win when the probability of losing is high (even more so than when the 
latter probability is low). Clearly, in such case the probabilities of objective outcomes 
such as winning and losing are directly involved in the description of subjectively 
relevant consequences such as ``I won (or lost) against all odds''. Such dependency 
introduces an element of interference between lotteries and consequences 
that cannot be easily accounted for in the classical decision-theoretic setting, 
which presumes state independence. 

An act is identified with a function $f : \mathcal{S} \rightarrow X$. $H$ is the set of all acts.

$\Delta (X)$ is the (nonempty, closed and convex) set of all probability
functions on $Z$ induced by lotteries in $X$.

$M$ is the set of all vectors $(p_s)_{s \in S}$, with $p_s \in \Delta (X)$.

For each $f \in H$ a corresponding risk profile $p^f \in M$ is
defined, for all $s \in \mathcal{S}$ and all $z^j \in Z,$ by $p^f_s
(z^j):= \left\langle z^j |V f_s \rangle^2 \right.$. 

As customary, we assume that the decision-maker's preferences are
characterized by a rational ({\emph{i.e.}}, complete and transitive)
preference ordering $\succsim$ on acts. Next, we proceed with the following assumptions, which mirror those in
Anscombe and Aumann (1963).

\begin{axiom}
  (Projective) There exists a finite orthonormal basis $Z:= \{z_1,...,z_m\}$, with $m \geq n$, such that any two acts
  $f, g \in H$ are indifferent if $p^f = p^g .$
\end{axiom}

In Anscombe and Aumann's setting, the above axiom is implicitly assumed to
hold with $Z \equiv \Omega$. Because of Axiom 6, preferences on acts can be equivalently
expressed as preferences on risk profiles. For all $p^f, p^g \in M$, we
stipulate that $p^f \succsim p^g$ if and only if $f \succsim g$.

\begin{axiom}
(Archimedean) If $p^f, p^g, p^h \in M$ are such that $p^f \succ p^g
  \succ p^h$, then there exist $a, b \in (0, 1)$ such that $a p^f + (1 - a)
  p^h \succ p^g \succ b p^f + (1 - b) p^h$.
\end{axiom}

\begin{axiom}
  (Independence) For all $p^f, p^g, p^h \in M$, and for all $a \in (0, 1]$,
  $p^f \succ p^g$ if and only if $a p^f + (1 - a) p^h \succ a p^g + (1 - a)
  p^h$.
\end{axiom}

\begin{axiom}
  (Non-degeneracy) There exist $p^f, p^g \in M$ such that $p^f$ $\succ p^g$.
\end{axiom}

\begin{axiom}
  (State independence) Let $s, t \in \mathcal{S}$ be non-null states, and let
  $p, q \in \Delta (X)$. Then, for any $p^f \in M$,

\smallskip
 
$(p^f_1,..., p^f_{s
  - 1}, p, p^f_{s + 1},..., p^f_n) \succ (p^f_1,..., p^f_{s
  - 1}, q, p^f_{s + 1},..., p^f_n)$ 

\smallskip

if, and only if,

\smallskip

$(p^f_1,...,
  p^f_{t - 1}, p, p^f_{t + 1},..., p^f_n) \succ (p^f_1,...,
  p^f_{t - 1}, q, p^f_{t + 1},..., p^f_n)$.
\end{axiom}

\begin{theorem}
  (Anscombe and Aumann) The preference relation $\succsim_{}$ fulfills Axioms
  $6 - 10$ if and only if there is a unique probability measure $\pi$ on
  $\mathcal{S}$ and a non-constant function $u : Z \rightarrow \mathbb{R}$ (unique up
  to positive affine rescaling) such that, for any $f, g \in H,$ $f \succsim
  g$ if, and only if,
  
  $\sum_{s \in \mathcal{S}} \pi (s) \sum_{z^i \in Z} p^f_s (z^i) u (z^i) \geq
  \sum_{s \in \mathcal{S}} \pi (s) \sum_{z^i \in Z} p^g_s (z^i)^{} u (z^i) .$
\end{theorem}

\smallskip

La Mura (2009) provides a projective generalization of Theorem 3.

\begin{theorem} (La Mura)
  The preference relation $\succsim_{}$ fulfills Axioms $6 - 10$ if and only
  if there is a unique probability measure $\pi$ on $\mathcal{S}$ and a
  symmetric $(n \times n)$ matrix $U$ with distinct eigenvalues such that,
  for any $f, g \in H,$ $f \succsim g$ if, and only if, 

$\sum_{s \in
  \mathcal{S}} \pi (s) f_s' U f_s \geq \sum_{s \in \mathcal{S}} \pi (s) g_s' U
  g_s .$
\end{theorem}

\section{Properties of the Representation} Our representation generalizes the
Anscombe-Aumann expected utility framework in two directions. First, subjective 
uncertainty and risk are treated as distinct notions. Specifically,
let us say that an act is $pure$ or $certain$ if it returns the same objective lottery in 
all states, and $mixed$ or $uncertain$ otherwise. 
While pure acts are naturally associated with risky decisions, in which the 
relevant frequencies are all known, mixed acts correspond to uncertain 
decisions, in which the decision-maker only has a subjective assessment of the true 
frequencies involved. Second, as we shall see, within this class of preferences
both Allais' and Ellsberg's paradoxes are accommodated.

In the context of Theorem 2, for any two distinct outcomes $\omega^i$ and $\omega^j$ 
let $e_{i, j}$ be the objective lottery returning each of the two outcomes with equal frequency. 
Observe that \[ U_{ij} = u (e_{i, j}) - ( \frac{1}{2} u (\omega^i) + \frac{1}{2} u (\omega^j)) . \]
It follows that the off-diagonal entry $U_{ij}$ in the payoff matrix can be
interpreted as the discount, or premium, attached to a symmetric, objective lottery over the two outcomes 
with respect its expected utility base-line, and hence as a measure of preference for risk versus uncertainty
along the specific dimension involving outcomes $\omega^i$ and $\omega^j .$
Let us say that a decision-maker is \textit{averse to uncertainty} if she always
weakly prefers $e_{i,j}$ to an equal subjective chance of $\omega^i$ or $\omega^j$. 
Then a decision-maker is averse to uncertainty if and only if $U$
is a Metzler matrix, \textit{i.e.}, has non-negative off-diagonal elements. 

Compared to existing generalizations of expected utility which avoid the Allais
or Ellsberg paradoxes, such as the ones in Machina (1982), Schmeidler (1989), or Chew, Epstein and Segal (1991),
among others, projective expected utility enjoys several advantages. Specifically, the representation is linear 
in the probabilities of states and consequences, hence remaining quite tractable, but can be 
nonlinear in the probabilities of the objective outcomes, hence allowing for portfolio effects. 
The axioms used to obtain the representation closely
mirror those introduced by Anscombe and Aumann (1963), which are widely regarded as appealing. Finally, as shown in 
sections 8 and 9, a compatible specification of the payoff matrix avoids both Allais' and Ellsberg's paradox.

\section{Example: Objective Uncertainty}

Figure 1 below presents several examples of indifference maps for pure
lotteries over three outcomes which can be obtained within our class of preferences for different
choices of $U$.

\begin{figure}
 \centering
 \includegraphics[width=7.5cm,height=6cm,bb=0 0 530 383]{inner-product4.jpg}
 % inner-product4.jpg: 530x383 pixel, 72dpi, 18.70x13.51 cm, bb=0 0 530 383
 \caption{Examples of indifference maps on the probability triangle}

\end{figure}

The first pattern (parallel straight lines) characterizes von Neumann -
Morgenstern expected utility. Within our class of representations, it
corresponds to the special case of a diagonal payoff matrix $U.$ All other
patterns are impossible within von Neumann - Morgenstern expected utility. 
Observe that, even though the payoff matrix is an object of relatively limited
algebraic complexity, the indifference curves can take a variety of different 
shapes: in particular, they do not need to be convex, or concave. Yet, since the indifference maps are generated
by a limited number of parameters (the entries in the payoff matrix), the type and
variety of preference patterns predicted by the model is also limited, and this in 
turn offers a basis for the empirical testability of the theory. 

\smallskip

The representation is sufficiently general to accommodate Allais' paradox from
section 3. In the context of the example in section 3, let $\{\omega^1, \omega^2, \omega^3
\}$ be the outcomes in which $4000,$ $3000,$ and $0$ dollars are won,
respectively. To accommodate Allais' paradox, assume that the non-diagonal 
elements of the payoff matrix increase with the difference between the 
corresponding diagonal payoffs. In the example below, the non-diagonal elements
are taken to be proportional to the fourth power of the difference.

\smallskip

$U = \begin{array}{c}
  \\
  \omega^1\\
 \omega^2\\
  \omega^3
\end{array} \begin{array}{c}
  \omega^1\\
  1.1\\
  0.00001\\
  0.14641
\end{array} \begin{array}{c}
  \omega^2\\
  0.00001\\
  1\\
  0.1
\end{array} \begin{array}{c}
  \omega^3\\
  0.14641\\
  0.1\\
  0
\end{array}$

\smallskip

The above formulation of the payoff matrix implies that, whenever the stakes 
involved are similar, the corresponding prospects are evaluated approximately 
at their (von Neumann-Morgenstern) expected utility values. By contrast, whenever 
the stakes are significantly different, the divergence from expected utility is also
significant.
Let the four lotteries $A,$ $B,$ $C,$ $D$ be defined, respectively, as the
following unit vectors in $\mathbb{R}^3_+$: $a := ( \sqrt{0.2}, 0, \sqrt{0.8})'$; $b := (0,
\sqrt{0.25}, \sqrt{0.75})'$; $c := ( \sqrt{0.8}, 0, \sqrt{0.2})'$; $d := (0, 1,
0)'$. Then lottery $A$ is preferred to $B,$ while $D$ is preferred to $C$, as

\smallskip

$u (a) = a' Ua = 0.33713,$

$u (b) = b' Ub = 0.3366,$

$u (c) = c' Uc = 0.99713,$

$u (d) = d' Ud = 1.$



\section{Example: Subjective Uncertainty} In the Ellsberg puzzle, suppose that
either all the non-red balls are green (\textit{i.e.}, 100 red, 200 green, 0 blue), or they
are all blue (100 red, 0 green, 200 blue), with equal subjective probability.
Further, suppose that there are just two objective outcomes, Win and
Lose. Then the following specification of the payoff matrix accommodates the
paradox.

\smallskip

$U = \begin{array}{c}
  \\
  \ $Win$\\
  \ $Lose$
\end{array} \begin{array}{c}
  \ $Win$\\
  1\\
  \alpha
\end{array} \begin{array}{c}
  \ $Lose$\\
  \alpha\\
  0
\end{array}$

\smallskip

As we shall see, if $\alpha = 0$ we are in the expected utility case, where
the decision-maker is indifferent between risk and uncertainty; when $\alpha >
0,$ risk is preferred to uncertainty; and when $\alpha < 0$, the
decision-maker prefers uncertainty to risk. In fact, let $\left\{ Urn1, Urn2 \right\}$ be the set of
possible states of nature, with uniform subjective probability, and let 
$r := ( \sqrt{1 / 3}, \sqrt{2 / 3})',$
$\overline{r} := ( \sqrt{2 / 3}, \sqrt{1 / 3})'$
be the lotteries associated to pure acts $R$ and $\overline{R},$ respectively.
Furthermore, let $w := ( 1, 0)',$ $l := (0, 1)'$
be the lotteries corresponding to a sure win ($W$) and a sure loss ($L$), respectively.
The mixed acts $G$ and $\overline{G} $ have projective expected utilities
given by

\smallskip

$u (G) = p (Urn1) u ( \overline{R})$+$p (Urn2) u (L)$,

$u ( \bar{G}) = p (Urn1) u (R)$+$p (Urn2) u (W)$.

\smallskip

One also has that
$u (W) = w' Uw = 1,$
$u (L) = l' Ul = 0$,
$u (R) = r' Ur = 1 / 3 + \alpha \sqrt{8} / 3,$
$u ( \overline{R}) = \overline{r}' U \overline{r} = 2 / 3 + \alpha \sqrt{8} /
3,$
and therefore

\smallskip

$u (G) = 1 / 3 + \alpha \sqrt{2} / 3$

$u ( \overline{G}) = 2 / 3 + \alpha \sqrt{2} / 3.$

\smallskip

It follows that, whenever $\alpha > 0,$ $R$ is preferred to $G$ and
$\overline{R}$ to $\bar{G}$, so the paradox is accommodated. When $\alpha <
0,$ the opposite pattern emerges: $G$ is preferred to $R$ and $\bar{G}$ to
$\overline{R}$. Finally, when $\alpha = 0$ the decision-maker is indifferent
between $R$ and $G,$ and between $\overline{R}$ and $\bar{G}$.

\section{Games with PEU preferences}

Within the class of
preferences characterized by Theorem 4, is it still true that every finite
game has a Nash equilibrium? If the payoff matrix $U$ is diagonal we are in
the classical case, so we know that any finite game has an equilibrium, which
moreover only involves objective risk (in our terms, this type of equilibrium
should be referred to as ``pure'', as it involves no subjective uncertainty).
For the general case, consider that $u (f)$ is still continuous and linear
with respect to the subjective beliefs $\pi$, while possibly nonlinear (but still polynomial) with
respect to risk. As we shall see, any finite game has an equilibrium even within this
larger class of preferences, although the equilibrium may not be pure (in our
sense): in general, an equilibrium will rest on a combination of objective
randomization and subjective uncertainty about the players' decisions.

A finite, strategic-form game with PEU preferences is a $n$-tuple $G:= (I,(A^i)_{i \in I},(U^i)_{i \in I})$, where $I$ is a finite set composed of $k$ players, $A^i$ represents the set of feasible actions for player $i$, and $U^i$ is player $i$'s payoff matrix on outcomes. An outcome is a complete assignment of actions $(a^1,...,a^k)$, identified with the natural basis in $\mathbb{R}^d$, where $d:=\prod_i |A^i|$. 
An act of player $i$ is a function $f^i$ from states $S^i$ to lotteries on $A^i$. Let $\mathcal{H}^i$ be the set of feasible acts for player $i$. We assume that $\mathcal{H}^i$ is a compact set which is also convex with respect to both objective and subjective mixing, in the following sense. 

\textbf{Assumption (Convexity):} For all $f^i, g^i \in \mathcal{H}^i$, and for all $a \in [0,1]$, there exist $h^i, l^i \in \mathcal{H}^i$ such that 

(i) $p^{h^i}_s= a p^{f^{i}}_s + (1-a) p^{g^{i}}_s$ for all $s$ (objective mixture) 

(ii) $u(l^i) = a u(f^i) + (1-a) u(g^i)$ (subjective mixture).

\smallskip

A profile of acts is a unitary vector $f:=(f^1,...,f^k)$, and the utility of a profile $f$ for player $i$ is $u^i(f)= E_\pi [f'U^i f]$.
An equilibrium is a profile $f^*$ such that $u^i(f^*_i,f^*_{-i}) \geq u^i(f_i,f^*_{-i})$ for all $f^i$.

\smallskip

\begin{theorem} 
Any finite game with PEU preferences has an equilibrium. 
\end{theorem}

\textbf{Proof:} First, let $b^i(f^{-i}):=\{ f^i: [u^i(f^i,f^{-i}) \geq  u^i(g^i,f^{-i})]( \forall g^i \in \mathcal{H}^i ) \}$. Next, let $\mathcal{H}:= \times_i \mathcal{H}^i$, and let $b: \mathcal{H} \rightarrow \mathcal{H}$ be the best response correspondence, defined by $b(f)= \times_i b^i(f^i)$. Observe that $b$ is a correspondence from a nonempty, convex, and compact set $\mathcal{H}$ to itself. In addition, $b$ is a nonempty- and convex-valued, upper hemi-continuous correspondence. It follows that the conditions of Kakutani's fixed point theorem are satisfied, and hence the best response correspondence $b$ has a fixed point: a profile of acts $f^*$ such that $f^* \in b(f^*)$. The acts at this fixed point constitute an equilibrium since by construction $f^{*i} \in b^i(f^{*-i})$ for all $i$. $\blacksquare$

\section{Games with PEU preferences: Examples}

Consider the following game:

$\begin{array}{c}
  Pl.1,Pl.2\\
  a^1\\
 b^1
\end{array} \begin{array}{c}
  a^2\\
  1.1,0\\
  0,1.1
\end{array} \begin{array}{c}
  b^2\\
  1,1.1\\
  1.1,1
\end{array}$

If the agents maximize vN-M expected utility, the unique mixed strategy equilibrium takes the following form: $(p, q) = (1/12,1/12) \approx (0.08333,0.08333),$
where p and q are the probabilities of playing $a^1$ and $a^2$, respectively. We shall call this equilibrium EU equilibrium.

If the players have the preferences we introduced to explain Allais’ paradox, what are the consequences on the equilibrium strategies of the players? In particular, does the equilibrium differ in the case of Allais’ agents? Consider the following game:

$\begin{array}{c}
  Pl.1,Pl.2\\
  a^1\\
 b^1
\end{array} \begin{array}{c}
  a^2\\
  \omega^1,\omega^3\\
  \omega^3,\omega^1
\end{array} \begin{array}{c}
  b^2\\
  \omega^2,\omega^1\\
  \omega^1,\omega^2
\end{array}$

where the payoff matrix of both players is given by $U$ in the Allais example. Observe that the payoff matrix $U$ describes a decision-maker who is strictly uncertainty averse, and hence if all players have payoffs matrices given by $U$ any equilibrium only involves pure acts. In this case, the unique equilibrium is given by $(p, q) = (0.17632,0.17632)$.

Compared to the case of EU preferences, in the game with PEU preferences $a^1$ and $a^2$ are played more often. Starting from the EU equilibrium, observe that strategy $a^2$ becomes more attractive for P2. To re-establish equilibrium $a^1$ must be played more often, but then to keep P1 indifferent between her two strategies $a^2$ must also be played more often. 

Next, consider the following very simple version of the Centipede game: P1 either quits (outcome; $\omega^2, \omega^3$) or passes, in which case P2 either quits (outcome; $\omega^3, \omega^1$), or passes (outcome; $\omega^1, \omega^2$). The payoff matrices of the two players are still given by $U$. 

Let $p$ denote the probability that P1 passes, and let $q$ be the probability that P2 passes given that P1 passed. Then the PEU for P1 and P2 are given, respectively, by

$((1-p)(1-q) ,p,(1-p)q ) U ((1-p)(1-q) ,p,(1-p)q )'$

$((1-p)q ,(1-p)(1-q),p ) U ((1-p)q ,(1-p)(1-q),p )'$.

In a subgame-perfect equilibrium, player 2 chooses $q$ so that it maximizes 

$(\sqrt{q} ,\sqrt{(1-q)},0 ) U (\sqrt{q} ,\sqrt{(1-q)},0 )'$, 

and hence will choose $q\approx 1$, in which case P1 will choose $p$ to maximize

$(0,\sqrt{p} ,\sqrt{(1-p)} ) U (0,\sqrt{p} ,\sqrt{(1-p)} )'$, 

and hence in equilibrium $p \approx 0.99029$.
It follows that the unique subgame-perfect equilibrium in this game involves a small probability of 
continuation. Clearly, in a longer version of the centipede those continuation probabilities would be amplified.

\section{References}

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Anscombe, F.J. and Aumann, R.J 1963. A Definition of Subjective 
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Bernoulli, D. 1738. Specimen theoriae novae de mensura sortis. In
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von Neumann, J. 1932. {\it Mathematische Grundlagen der
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von Neumann, J., and Morgenstern, O. 1944. {\it Theory of Games and
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