Dynamic Gambling under Loss Aversion
Yair Antler
Tel Aviv University
January 2015 (preliminary and incomplete)
Abstract
We study a problem of dynamic gambling in which a loss-averse
gambler faces an in…nite sequence of unfair lotteries. We …nd that the
optimal gambling strategy is the following stopping rule: the gambler
stops gambling after accumulating small net gains or large net losses.
In a static environment, our gambler has a preference for left-skewed
lotteries. These two …ndings are in contrast to the common wisdom
that gamblers with prospect theory preferences have a preference for
right-skewed lotteries. Furthermore, we show that there always exists
a sequence of unfair binary lotteries such that the gambler is willing
to participate in some of them. We apply our results to a problem of
casino pricing and demonstrate how it may be pro…table for a casino
to provide inducements in order to attract gamblers into the casino.
1
Introduction
Attempts to understand gambling behavior date back to Friedman and Savage (1948) who introduced a non-concave segment into the utility function
in order to settle expected utility theory and gambling. With similar goals,
Markowitz’s (1952) seminal paper set the foundations for preferences that are
de…ned over gains and losses instead of …nal assets. Kahneman and Tversky
This paper has bene…ted from ERC grant no. 230251. I am grateful to Rani Spiegler
for his valuable guidance. I thank Eddie Dekel, K…r Eliaz, and Asher Wolinsky for helpful
conversations and comments. All remaining errors are mine. Correspondence: Yair Antler,
The Eitan Berglas School of Economics, Tel Aviv University. E-mail: [email protected].
1
(1979, 1992) introduced a theory in which agents evaluate risk using a value
function that is de…ned over gains and losses (concave over gains and convex
over losses) and that is kinked at the origin. In their theory, agents distort
probabilities by applying a weighting function and tend to give relatively
high weight to low probabilities.
In a dynamic context, Barberis (2012) used Kahneman and Tversky’s
(1992) cumulative prospect theory to demonstrate numerically that for a wide
range of preference parameters, a prospect theory agent is willing to gamble
in a casino even if the casino o¤ers only bets with no skewness and with zero
or slightly negative expected value. In his seminal example, the gambler’s
strategy induces a right-skewed binary lottery. Roughly speaking, lotteries
that are skewed to the right are attractive to gamblers who overweight low
probabilities. Ebert and Strack (2015) generalize Barberis’ example. In
a dynamic context, they show that the same e¤ect makes a dynamically
inconsistent naive1 gambler go bankrupt since there always exists a rightskewed unfair lottery that is attractive to the gambler. In both papers, the
driving force behind the preference for right-skewed lotteries in both of these
papers is probability distortion and not loss aversion.
In the present paper, we study the behavior of a loss-averse gambler.
Loss-aversion is the tendency of people to evaluate changes or di¤erences
rather than absolute magnitudes and to dislike losses more than they like
comparably sized gains. The gambler in our work is dynamically consistent
(his reference wealth does not change over time) and he has the right perception of probabilities (he does not distort probabilities). Faced with an
in…nite sequence of identical unfair lotteries, each of which pays k ( k) dollars with probability p (1 p), our gambler has to decide in which (if any) of
the lotteries to participate. his preferences over gains and losses are represented by a utility function that is concave for gains, convex for losses, and
has a kink at his reference wealth. We refer to the gambler’s wealth before
he enters the casino as his "reference wealth" and we interpret his decision
to participate in gambling activity (at the reference wealth) as a decision to
enter the casino.
It turns out that loss aversion without probability distortion leads to a
behavior that is di¤erent from the one described in Barberis (2012) and Ebert
and Strack (2014). In particular, if our gambler chooses to enter the casino,
1
The gambler is naive in the sense that he is unaware that his reference wealth is
updated after each lottery.
2
his optimal gambling strategy is to stop gambling after accumulating a small
number of net gains or a larger number of net losses. Surprisingly, we show
that a casino can always choose an unfair lottery (p; k) such that the gambler
will enter the casino. In a static context, our results imply that loss averse
gamblers have a preference for left-skewed lotteries.
Since the gambler has a convex segment in his utility function, it is not
very surprising that he is willing to accept unfair lotteries after he accumulates some net losses. The striking result is that the loss-averse gambler is
willing to accept unfair lotteries even at his reference-wealth (that is, without
accumulating net losses). Let us try and understand this e¤ect. Consider
the following utility function, originally proposed by Kahneman and Tversky
(1992):
(x r)
f or x r
v (x) =
(1)
(r x)
f or x < r
where x is the gambler’s wealth, r is his reference wealth, > 1 is a lossaversion parameter and, < 1 is the curvature of the utility function. Recall
that the utility function given in (1) is "S-shaped." This means that there
are wealth-levels x < r such that
p (r x + k) (1 p) (r x k) >
(r x) . That is, wealth levels at which participation in one lottery is
attractive to the gambler. This means that with a wealth of x, the value
of executing the optimal gambling strategy from this point onwards, V (x)
is strictly greater than v (x) =
(r x) . Therefore, it is possible that
pk
(1 p) k < 0 and pk + (1 p) V ( k) > 0. That is, although one
lottery is not attractive to the gambler to participate in when he is at x = r,
the opportunity to gamble at the convex segment of his utility function makes
gambling attractive at x = r.
To illustrate this e¤ect, we set r = 0; k = 1; = 1:1; p = 0:49, and
= 0:5. The utility that is derived from participation in one lottery at the
gambler’s reference wealth is 0:49 (1 0) + (0 1) 0:51 1:1 = 0:071. The
utility that is derived from one lottery at x = 1 is 0:51 1:1 20:5 =
0:793 > v ( 1) = 1:1. Note that the value of gambling once in case
of winning the …rst lottery, or twice in case of losing in the …rst lottery, is
0:49 1:1 0:51 0:793 = 0:045 > v (0). That is, the option to gamble once
again and break even increases the value of x = 1 and makes gambling
attractive at x = r.
In the last part of the paper we apply our results to a problem of monopolistic casino pricing. Our casino o¤ers an in…nite sequence of unfair
3
lotteries that take place at regular intervals. The casino controls the stakes
of these lotteries but it cannot change the lotteries’odds (this assumption is
supported by the fact that gambling games are canonical lotteries with given
probabilities. For example, the famous Roulette game ’Red or Black’ is a
18 19
lottery that pays x ( x) dollars with probability 38
( 37 )).
It is well-known that casinos provide gambling inducements in the form
of complimentaries (free drinks, food, hotel rooms, etc.). Motivated by this
phenomenon, our casino has the ability to o¤er gambling inducements in order to attract gamblers. A gambler who chooses to participate in at least one
lottery is entitled to an inducement that is o¤ered by the casino. We think
of the inducement as a complimentary that does not change the gambler’s
reference wealth. We show that if the casino is allowed to provide inducements, it may improve its pro…ts as it can attract gamblers who otherwise
would not enter the casino.
The interesting fact about this e¤ect is that even if the inducement’s
cost to the casino is identical to the utility that the gambler derives from
the inducement, it is still pro…table for the casino to provide it in many
cases. Also, we show that when the casino chooses to provide an inducement,
it increases with the stakes of the lotteries. This result is consistent with
the fact that gamblers who bet on greater stakes, are usually o¤ered higher
inducements than those o¤ered to regular gamblers.
The paper proceeds as follows. In Section 2 we present the model. In Section 3 we characterize the gambler’s behavior. Section 4 studies the casino’s
optimal pricing scheme and Section 5 concludes.
2
The model
We consider a gambler who faces an in…nite sequence of identical lotteries.
At each period t 2 f1; :::1g, the gambler faces an unfair lottery that o¤ers
a prize of k ( k) dollars with probability p (1 p). The lotteries take place
at the end of each period. Let at 2 fY; N g denote the gambler’s periodic
decision whether to participate in a lottery at date t. Let xt denote the
gambler’s wealth at the beginning of date t and let r denote his reference
wealth (his initial wealth before he enters the casino). We normalize r = 0
and refer to xt as the gambler’s net gains/losses at date t. Let ht denote the
history (x0 ; a0 ; :::; at 1 ; xt ) at date t. A gambling plan speci…es for each
period t 2 f1; :::1g which decision at to choose as a function of the history
4
ht . Note that, for all t, xt 2 f tk; (1 t) k; :::; tkg. A gambling plan is
called stationary if, for each t 2 f1; :::1g, the decision at period t depends
only on xt . It is assumed that in each period t there is a probability 1
of
an immediate end to the game.
Throughout the analysis we assume that the gambler is dynamically consistent. That is, his reference wealth does not change during his visit in the
casino. This assumption can be replaced by an assumption that, in each
period t 2 f1; :::; 1g, with some probability q < 1, the gambler’s reference
wealth is updated to xt . If q is not too high, the qualitative results in the
paper do not change. Also, it is assumed that the gambler is fully rational in
the sense that he predicts his future behavior perfectly. Finally, we assume
that the gambler has a budget w such that, if he looses w dollars, he is ruined
and must stop gambling.
We assume that the gambler’s preferences over gains and losses are represented by a utility function v : R ! R that is concave for gains, convex for
losses and is kinked at the origin (at the reference wealth). Formally, it is
assumed that v is di¤erentiable everywhere (except at the reference wealth),
v (0) = 0, v 0 (x) > 0, v 00 (x) > 0 for x < 0 and v 00 (x) < 0 for x > 0. It is also
assumed that v (x) < v ( x) for all x > 0 (i.e., the gambler is loss-averse).
For example, the utility function given in (1) is consistent with our assumptions for > 1, and < 1. Note that v is de…ned over …nal outcomes (that
is, …nal net gains/losses at the end of the game). We de…ne V ; (x) to be the
expected continuation value for the gambler when he has gained x dollars
and plans to follow the gambling plan .
3
The Gambler’s Behavior
We focus on stationary strategies. The following lemma establishes that this
assumption is without loss of generality.
Lemma 1 There exists a stationary gambling plan
which is optimal.
The lemma is a straightforward application of Blackwell’s (1965) theorem,
therefore, its proof is omitted. Throughout the analysis we assume that
whenever the gambler is indi¤erent between participating in a lottery or not
at some wealth-level, then he participates in it. This assumption pins down a
unique optimal gambling plan. From now on, we interchangeably refer to x as
5
the gambler’s gains level and as the state. In what follows we characterize the
gambler’s optimal stationary gambling plan given a periodic lottery (k; p).
w
. The following
Denote with lw the highest integer l such that l
k
proposition relates to the strategy of a gambler who decides to enter the
casino. It establishes that if the gambler enters the casino (that is, if (0) =
Y ), he stops gambling when he accumulates l
lw net losses or u < l net
wins.
Proposition 1 Suppose that is an optimal stationary strategy. If (0) =
Y , then there exist two integers u 1 and lw l > u such that
( lk) =
(uk) = N and (xk) = Y for all x 2 f(1 l) k; :::; (u 1) kg.
Proof. First, we show that it cannot be that
(x) = Y for each x 2
f(1 lw ) k; :::; k; 0; k; :::g. Suppose that this is the case. Note that the gambler’s value is increasing in p. Set p = 21 . Suppose that the gambler cannot be
ruined. Since v is convex for x < 0, 12 v ((x 1) k) + 21 v ((x + 1) k) > v (xk)
for x 2 f:::; 1g. By the de…nition of the value function, 21 v ((x 1) k) +
1
V ; ((x + 1) k) > v (xk) for all x 2 f:::; 1g. It follows that a strategy 0
2
such that 0 (x) = Y for all x 2 f:::; k; 0; k; :::g, does strictly better than
in the unconstrained problem. Note that 0 induces a symmetric spread
around 0. By loss-aversion, the value that 0 induces in the unconstrained
problem at wealth 0 is negative, namely V 0 ; (0) < 0. It follows that in the
constrained problem, V ; (0) < 0 = v (0).
By now we have established that if
(0) = Y , then there exists an
integer u > 0 such that
(uk) = N . Since the gambler cannot lose more
than w dollars, there is an integer l 2 flw ; :::; 1g. such that ( lk) = N .
It is left to show that l > u. Suppose not and …x p = 21 . Note that if l = u,
then induces a symmetric spread around 0. This implies that V ; (0) < 0.
Suppose that l < u. Set p = 12 and consider a gambler that cannot be ruined.
By an argument similar to the one used previously, a strategy in which the
gambler stops after u = l net wins or losses does strictly better than
in
the unconstrained problem. It follows that V ; (0) < 0, a contradiction.
It follows from proposition 1 that if the gambler enters the casino (that
is, if
(0) = Y ), he stops gambling after accumulating u net wins or l net
losses, where l > u. These results are opposed to Barberis (2012) and Ebert
and Strack’s (2014) results. In their work, because of the relatively large
weight the gambler puts on low probabilities, he plans to stop gambling after
large gain - an event that happens with a very low probability, or a small
loss. In their work, this e¤ect dominates the e¤ect of loss-aversion.
6
Proposition 1 makes strong use in the assumption that the gambler is
constrained (he is ruined in case he losses w dollars). In proposition 2 we
relax this assumption by assuming that the gambler cannot be ruined (that is,
he is allowed to continue gambling regardless of his net losses). We show that
if v is not bounded from below, the result of proposition 1 holds. Since the
proof is similar to the proof of proposition 1, it is relegated to the appendix.
Proposition 2 Suppose that the gambler cannot be ruined and that v is not
bounded from below. If
(0) = Y , then there exist two integers u
1
and l > u such that
( lk) =
(uk) = N and
(xk) = Y for all
x 2 f(1 l) k; :::; (u 1) kg.
Note that propositions 1 and 2 do not address the gambler’s strategy o¤
the equilibrium path. Speci…cally, it is impossible to characterize the gambler’s behavior in wealth levels x 2 f lw k; :::; (l + 1) kg without making
any
additional assumptions on v. For the next proposition we assume that
v 00 (x)
is decreasing in x for x < 0. Roughly speaking, it is assumed the
v 0 (x)
the gambler becomes less risk-loving as his wealth moves further from his
references-wealth. This assumption is satis…ed, for example, by the utility
function given in (1).
Proposition 3 Suppose that
(0) = Y . If v exhibits decreasing risk-aversion
00
(x)
decreases in x for x < 0), then (x) = Y if and
for x < 0 (that is, if uu0 (x)
only if x 2 f(1 l) k; :::; (u 1) kg, where l and u are two integers such that
l > u 0.
This proof makes a strong use of a claim made by Samuelson (1963): If
at each wealth level within a range the expected utility of a certain bet is
worse than abstention, then no sequence of such independent ventures have
a favorable expected utility.
Proof. We know from proposition 1 that there exists an integer u
0
such that
(ku) = N . Since v is concave for x > 0, pv ((x + 1) k) +
(1 p) v ((x 1) k) < v (xk) for x 2 fk; 2k; :::g. It follows from Samuelson
(1963) that if (x) = N for x 2 f0; k; :::g, then it must be that (x0 ) = N
for each x0 2 fx; x + k; :::g. Suppose that lw > l. If
( lk) = N , it
must be that pv ( (l 1) k) + (1 p) v ( (l + 1) k) < v ( lk), for otherwise
00 (x)
pV ; ( (l 1) k)+(1 p) V ; ( (l + 1) k) v (lk). Since uu0 (x)
decreases
7
in x for x < 0, one lottery is not desirable for any x 2 f:::; lkg. Therefore,
we can apply Samuelson’s claim once again and get that
(x) = N for all
x 2 flw k; :::; lkg. The argument that l > u is omitted since it is similar to
the same argument made in proposition 1.
So far, we have left open the question whether gamblers enter the casino
in the …rst place. That is, whether (0) = Y . Proposition 4 addresses this
question. It shows that in the = 1 limit, there exist a probability p < 12
and stakes k, such that (0) = Y if, and only if, condition (1) holds.
Condition 1 There exists a budget w0
uw0
l
l
v
l!1 u
lim
w such that:
>
v ( w0 )
Condition (1) is satis…ed, for example, by the utility function given in
(1). The RHS of condition (1) represents the possible utility loss from a
stopping rule according to which the gambler stops gambling at a net loss of
w0 dollars. On the LHS of condition (1) one can …nd the possible utility gain
from the same stopping rule weighted by the relative likelihood of stooping
after u net wins in the p = 12 limit.
Proposition 4 Fix w and consider the = 1 limit. There exist a probability
p < 21 and stakes k such that (0) = Y if, and only if, condition (1) holds.
Proof. By proposition 1, (0) = Y if, and only if, there is a stopping rule
( kl; ku) that induces a value V ; (0)
0. Consider the = 1 limit.
Suppose that there exists a stopping rule
such that the gambler stops
gambling at x 2 fuk; lkg. Then,
lim V
!1
;
(0)
:
1 p
p
=
1 p
p
We take the p =
(2)
= V (0)
1
2
0
l
1
l+u
1
B
v (uk) + @1
1 p
p
1 p
p
l
l+u
1
1 C
A v ( lk) > 0
1
limit:
lim1 V (0) =
p! 2
l
u
v (uk) +
v ( lk) > 0
l+u
l+u
8
(3)
If inequality (3) holds, then the claim is proven since one can always set the
appropriate p < 21 . Denote w0 := lk and recall that w0 w. Inequality (3)
holds if, and only if:
l
uw0
u
v
+
v ( w0 )
l+u
l
l+u
l
uw0
u
v
+ v ( w0 ) > 0
=
l+u u
l
(4)
Fix an arbitrary integer g. For any …xed k, there is a probability p su¢ ciently
close to 0:5, such that it is optimal to gamble at x 2 f gk; :::; kg. Since
we are free to choose a p su¢ ciently close to 21 , we can decrease k and
increase l while keeping the product lk …xed (and not violating the gambler’s
0
budget constraint). By Jensen’s inequality, lv uwl
is strictly increasing
in l since v is concave in x > 0. It follows that there exists an integer l
0
such that ul v uwl + v ( w0 ) > 0 and inequality (4) holds if, and only if,
> uv ( w).
liml!1 lv uw
l
Proposition 4 implies that as long as condition (1) holds, one can always
o¤er the gambler a sequence of unfair lotteries such that he would enter the
casino. The driving force behind this result is that the convex segment to
the left of the reference-wealth creates a "break-even" e¤ect (Johnson and
Thaler (1990)). Lotteries are attractive to the gambler in that segment, and
therefore, the opportunity to continue gambling increases the option value of
a loss (the continuation value of a loss is strictly greater than the utility that
is derived by the same loss). It follows that although pv (k)+(1 p) v ( k) <
v (0), it may very well be that pv (k) + (1 p) V ; ( k) > v (0).
As an illustration, consider a gambler that his preferences are represented
by (1) and set = 1:5; = 0:5; w = 10, k = 2. Let us consider the famous
and study the gambler’s strategy. Since
’Red or Black’game, in which p = 18
37
18
19 0:5
0:5
0:5
10
6
>
8
,
(
8)
=
Y . Since v exhibits decreasing risk37
37
aversion, (x) = Y for x 2 f 6; 4; 2g. Suppose that (2) = N , (0) =
4
5
4
(1 p) 1
( 1pp ) ( 1pp )
Y . Then, lim !1 V ; (0) = 1 p p 5 20:5 1:5
(10)0:5 > 0. It
5
( p ) 1
( 1pp ) 1
follows that there is a stopping rule that induces a strictly positive expected
utility for the gambler. Therefore, the gambler enters the casino. Note that
one lottery is not attractive to the gambler at his reference wealth since
18 0:5
19 0:5
2
1:5 37
2 < 0.
37
In a static context, proposition 4 implies that loss-averse gamblers have a
taste for left skewed lotteries. Speci…cally, it implies that as long as condition
9
(1) holds, there is an unfair left-skewed lottery that is acceptable for the
4
(1 p) 1
gambler. For example, consider the parameters above. Let q := 1 p p 5 and
( p ) 1
note that the lottery (2; q; 10; 1 q) is a binary unfair left-skewed lottery
that is acceptable by the gambler.
4
Casino Pricing
In this section we apply our previous results and study a problem of casino
pricing. The analysis in this section is for the = 1 limit. Motivated by the
observation that most casino games are canonical games with strict sets of
rules, we assume that the casino cannot control p. Rather, our casino has two
choice variables, the betting stakes k and a gambling inducement g. Formally,
given p, the casino commits to a contract hk; gi such that (k; k; p; 1 p) is
a lottery which is played over and over at times t = 1; 2; :::; 1 and g is an
inducement that is given to gamblers who choose to accept the contract.
A gambler who accepts a contract must participate in the lottery played
at date 1 and is allowed to play at times 2; :::; 1 if he wishes (but he is not
obligated to do so). Upon accepting the contract he is also entitled to an
inducement g. Our interpretation of a contract is of a gambling room in
which the lottery (k; k; p; 1 p) is played and drinks worth g are served.
Accepting such a contract is interpreted as entering the gambling room.
There are two ways to consider the inducement’s e¤ect on the gambler’s
well-being. One option is to think of g as a non-monetary component such
that the utility of a gambler who gets an inducement of g is v (x) + g. The
second option is to think of g as a monetary inducement such that the utility
of a gambler who gets an inducement of g is v (x + g). It turns out that our
qualitative result in this section does not depend on the way that the gambler
accounts for g as long as we assume that the inducement does not a¤ect the
gambler’s reference-wealth. We …nd this assumption easier to justify when
g is a non-monetary inducement. Formally, we assume that the gambler’s
preferences are quasi-linear such that the utility of a gambler with …nal wealth
x and inducement g is v (x) + g.2 Moreover, we assume that v is homogenous
of degree < 1. That is, for all x > 0, v (kx) = k v (x).
Illustration. In the previous section we have showed that a gambler might
want to gamble at x = 0. Here we consider a case in which the gambler …nds
2
Recall that the reference wealth is normalized to 0.
10
gambling unattractive at x = 0 (he rejects any contract hk; 0i. We show that
there exists an inducement g > 0 such that the gambler is willing to accept
the contract hk; gi and the casino makes an expected pro…t that is greater
18
,
than g. Consider the functional form given in (1). Suppose that p = 37
0
w = 10, = 0:5 and = 2. Consider the strategy
in which the gambler
stops at x 2 f 10; 0; 2g. That is, the gambler stops when he is either ruined,
wins the …rst lottery, or gets back to his reference wealth. In case the gambler
accepts a contract h2; 0i, the strategy 0 induces the greatest expected value
for him.3 However, his expected value under 0 is:
0
1
4
3
lim V
!1
0;
18 0:5
2
(0) =
37
2 (1
B
p) @
1 p
p
1 p
p
1 p
p
4
1
C
0:5
A (10) =
0:191
This means that with no inducement, the gambler will not enter the casino.
On the other hand, this strategy induces an expected pro…t of 0:416 to the
casino. Therefore, there is an inducement g 2 (0:191; 0:416) such that the
gambler is willing to accept the contract h2; gi and the casino makes a strictly
positive net expected pro…t. This follows from the fact that in the convex
segment of his utility function, both the gambler and the casino bene…t from
his gambling. It is worthwhile for the casino to pay for the chance that
the gambler will lose and reach the convex segment of his utility function.
Relative to a case in which the gambler plays only once, the casino makes
a greater pro…t and has to pay a lower inducement in order to attract the
gambler to enter the casino.
We now present the main result of this section. We show that the betting
stakes k are increasing in the gambler’s budget w. Moreover, the optimal
inducement g increases with the betting stakes. Throughout the analysis we
denote the casino’s optimal contract with hk ; g i.
Proposition 5 Suppose that v is homogenous of degree < 1. If the casino
makes a strictly positive expected pro…t, then the gambling stakes are increasing in w and the inducement g is increasing in the gambling stakes.
Proof. First, we make a distinction between two cases: either there exists a
contract hk; 0i that is acceptable for the gambler, or not.
3
In order to verify that, simply note that by decreasing risk aversion,
x 2 f 8; :::; 2g and that
(0) = N regardless of the upper-bound.
11
(x) = Y for
Case 1 : there exists no acceptable contact hk; 0i.
Since the casino can make a strictly positive pro…t, it must o¤er an acceptable contract hk; gi. Let us consider the optimal gambling strategy of
gambler who accepts a contract hk; gi. Since hk; 0i is unacceptable, the gambler stops gambling whenever he reaches state 0 (after participating in the
…rst lottery). It follows from the arguments made in proposition 3 that he
also stops gambling if he wins the …rst lottery. By proposition 1, there is an
w
such that the gambler stops gambling if he accumulates l net
integer l
k
losses. Denote it with l (k ). The value that hk ; g i induces for the gambler
is 0, for otherwise, the casino can decrease g by a small " > 0 and increase
its expected pro…t. It follows that:
0
1
l(k ) 1
1 p
1C
p
B
g = (1 p) @1
(5)
A v ( l (k ) k ) pv (k )
l(k )
1 p
1
p
The casino’s expected pro…t from gambling is:
0
1
l(k ) 1
1 p
1C
p
B
(1 p) @1
A l (k ) k
l(k )
1 p
1
p
pk
(6)
Suppose that k < l(kw ) . Fix an arbitrary small " > 0. Since v is homogenous of degree < 1, the casino can increase its expected pro…t by o¤ering
the contract h(1 + ") k ; (1 + ") g i which is acceptable for the gambler and
multiplies the casino gains by 1 + ". Therefore, l (k ) k = w. Let l1 denote
the number of accumulated net losses after which the gambler stops gambling in the w = 1 limit. The casino’s problem is equivalent to choosing
l 2 f1; :::; l1 g in order to maximize its expected pro…t from gambling (7)
minus the inducement given in (8).
1
0
l 1
1 p
1C
p
w
B
(7)
(1 p) @1
Aw p
l
l
1 p
1
p
0
1
l 1
1 p
1
p
w
B
C
(1 p) @1
(8)
A v ( w) pv
l
l
1 p
1
p
12
When l 2 f1; :::; l1 1g is increased by 1, the (negative) change in the
casino’s expected pro…t from gambling activity is given in (9) and the (negative) change in the inducement is given in (10).
2
0
3
1
l
l 1
1 p
1 p
1
1C
p
p
p 7
6
B
w 4(1 p) @
(9)
5
A
l+1
l
l (l + 1)
1 p
1 p
1
1
p
p
2
0
1
l
l 1
1 p
1 p
1
1C
p
p
1
1
6
B
v
w 4 (1 p) @
A v ( 1) p v
l+1
l
l
(l + 1)
1 p
1 p
1
1
p
p
(10)
If expression (9) (the decrease in the gains) minus expression (10) (the decrease in the inducement) is positive, it is not bene…cial for the casino to
increase l by 1. Since an unconstrained gambler would like to participate in
l1 lotteries, expression (10) is positive. Consider a gambler with w = w0 .
Suppose that for this gambler, it is not bene…cial for the casino to increase
l by 1. This means that expression (9) is positive. It follows that it is not
bene…cial for the casino to increase l by 1 for any gambler with w > w0 .
Therefore, l (k ) (k ) is weakly decreasing (increasing) in w.
It is left to show that g is increasing in k . Recall that the net value of
the contract for the gambler is 0. Fix l. Multiplying k by 1 + " multiplies
the gambler’s negative value from gambling by (1 + ") , and therefore, g is
increasing in w for a …xed l. By the previous arguments, if l changes when
w increases, then it decreases. It follows that the two changes a¤ect g in
the same direction. Therefore, the gambler’s utility from gambling decreases
such that the inducement g must increase to keep the gambler indi¤erent
between accepting and rejecting the contract.
Case 2 : there exists an acceptable contract hk; 0i.
In this case, it is possible that g = 0. First, we show that, in this
case, k increases in w. Suppose that a contract hk ; 0i is accepted by the
gambler. The gambler stops gambling after accumulating u (k ) 1 wins or
l (k )
min kw ; l1 losses. By the previous arguments, l (k ) = kw . The
casino’s expected pro…t as a function of l (k ) is given by:
20
1 0
1
3
l(k )
l(k )
1 p
1 p
1 C B
1 C u (k ) 7
p
p
6B
w 4@1
5
A @
A
l(k )+u(k )
l(k )+u(k )
l (k )
1 p
1 p
1
1
p
p
13
3
7
5
Note that the casino’s decision on l is independent of w. Therefore, k = l(kw )
increases in w. The case of g > 0 is follows from case 1. It is left to show
that if g > 0 is optimal given w, g = 0 cannot be optimal given w0 > w.
Suppose that the casino o¤ers the contract hk ; g i with g > 0 for a given
budget w and that it o¤ers a contract hk ; 0i for a budget w0 > w. Denote the
casino’s induced pro…t from the contract hk; gi given w as (hk; gi ; w). Note
0
w
k ; 0 ; w = (hk ; 0i ; w0 ).
that if hk ; 0i is optimal given w0 , then ww
w0
0
w0
w0
Also, for a small " > 0, w (hk ; g i ; w) <
k ; ww g
" ; w0 . However,
w
w
w0
w0
k ;0 ;w
(hk ; g i ; w) and
k ; wg
" ; w0
(hk ; 0i ; w0 ),
w0
w
a contradiction.
We learn from proposition 5 that a less constrained gambler bets on
greater stakes. Moreover, "high-rollers", namely, gamblers who bet on greater
stakes, receive greater inducements. This result is in line with o¤ers and promotions made in real world casinos.
5
Concluding remarks
We have characterized the gambling behavior of a loss-averse gambler and
show that the optimal gambling strategy for such a gambler is a stopping
rule: the gambler stops gambling after accumulating small net gains or large
net losses. The driving force behind this behavior is that the gambler’s option
to "break even" increases the option value of a loss. This is consistent with
the "break even-e¤ects" described by Johnson and Thaler (1990) and with
the "disposition e¤ect" (Hersh and Statman (1985)).
The results obtained in this paper about the optimal gambling strategy
under loss-aversion imply that the results obtained by Barberis (2012) and
Ebert and Strack (2014) are not due to loss-aversion, rather the preference
that they …nd for right-skewed lotteries follows from the probability distortion. Namely, from the fact that in their work, gamblers overweight events
that happen with a relatively low probability. Their work implies that the
probability distortion’s e¤ect dominates the loss-aversion e¤ect. Although
their (our) gambler has a taste for right-skewed (left-skewed) lotteries, in
Ebert and Strack’s (2014) dynamic setup, the naive gambler’s behavior is
similar between the two models. In Ebert and Strack’s (2014) setup, there
is always a right-skewed lottery that is attractive for the gambler, until he
is ruined. Proposition 4 implies that there is always a stopping rule that
14
induces a left-skewed lottery that is attractive for the gambler to participate
in (until he is ruined).
References
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[3] Blackwell, David (1965): "Discounted Dynamic Programming." The
Annals of Mathematical Statistics 36, no. 1, 226–235
[4] Coolidge, Julian L. (1909): "The Gambler’s Ruin." Annals of Mathematics, Second Series,10 , no. 4, pp. 181-192
[5] Ebert, Sebastian and Strack, Philipp (2015): "Until the Bitter End: On
Prospect Theory in the Dynamic Context." American Economic Review:
forthcoming.
[6] Friedman, Milton, and Leonard J. Savage (1948): "The utility analysis
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15
[11] Samuelson, Paul A. (1963): "Risk and uncertainty: A fallacy of large
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6
Appendix
Proof of proposition 2
First, we show that it cannot be that (x) = Y for each x 2 f:::; k; 0; k; :::g.
Suppose not. Note that the gambler’s value is increasing in p. Set p = 12 .
Recall that v (x) < v ( x) for all x > 0 and
results in a symmetric
spread around 0. It follows that V (0) ; < 0 = v (0), a contradiction to the
optimality of .
We now show that it cannot be that
(x) = Y for each x 2 f0; k; :::g.
Suppose not. By the previous argument there exists a y 2 f:::; 2k; kg
such that (y) = N . Set p = 21 . Let 0 be a strategy such that 0 (x) = Y
for each x 2 f:::; k; 0; k; :::g. Recall that V (0) 0 ; < 0. Since v is concave
for x < 0, we can use the same arguments made in proposition 1 and show
that for p = 21 , V (0) 0 ; > V (0) ; . However, the gambler’s value decreases
in p, so for p < 21 , we get a contradiction since V (0) ; < 0 = v (0).
We now show that it cannot be that (x) = Y for each x 2 f:::; 2k; kg.
is optimal, V ; (x)
V ; 0 (x) for
Suppose not. Fix p < 21 . Since
every 0 > and x 2 f:::; k; 0; k; :::g. Consider the highest wealth x0 2
f:::; 2k; k; :::g for which (x0 ) = Y . In order to get a contradiction, it is
su¢ cient to show that lim !1 V ; (x0 ) < v (x0 ). Fix an arbitrary small " > 0.
It follows directly from the well-known "gambler’s ruin" problem (Coolidge,
1909) that for each state z 2 f:::; x0 2k; x0 kg:
lim Pr fThe game ends at a state x < zg >
!1
1 2p
1 p
"
By assumption, v (x) is not bounded from below. Therefore, lim !1 V ; (x0 ) <
v (x0 ). The fact that l > u follows from the same argument mad in Proposition 1.
16