Journal of Economic Behaviorand Organization
ETSEVIER
JOURNALOF
Economii Behavior
& Organization
Vol. 28 (1995) 417-442
The strategic implications of sunk costs:
A behavioral perspective
Roth Parayre
’
Southern Methodist Universiiy. Cox School of Business, Dallas 7X 75275, USA
Received28 June 1993;revised 30 November 1994
Abstract
This paper examines some of the strategic implications of the sunk cost phenomenon in
sequential allocation decisions. Drawing from psychology and behavioral decision theory,
we first present a taxonomy of possible causes for the ‘sunk cost effect’, the tendency of
many managers to throw good money after bad. We then present the analysis of some
implications of this behavior in strategic situations. A two-period strategic game is
developed and analyzed to derive optimal allocations as a function of one player’s sunk cost
behavior. We establish when this behavior can be used as a successful precommitment
strategy by the sunk cost player, and when it is exploitable by an opponent. Welfare
implications are also explored. The sunk cost effect constitutes a form of strategic
preference
manipulation, in which a credible preference change can be induced to provide a
strategic advantage to one of the players.
JEL classification: C72; DSl; D92
Keywords: Sunk cost; Escalation of commitment; Preference change; Duopoly; Signailing game
‘[People] are irrational, that’s all there is to that! Their heads are full of cotton,
hay and rags!’ - Alan Jay Lerner, My fair lady.
’ The comments of James Brander, Kenneth MacCrimmon, Donald Wehrung, Colin &meter, two
anonymous referees and the Editor are gratefully acknowledged. The usual disclaimer applies. Detailed
mathematical proofs are available from the author, upon request.
0167-2681/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved
SSDI 0167-2681(95)00045-3
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R. Parayre / J. of Economic Behavior & Org. 28 (19951417-442
. . . between the thin theory of the rational and the full theory of the true and
the good there is room and need for a broad theory of the rational.’ - Jon Elster,
1983, Sour grapes: Studies in the subversion of rationality.
1. Introduction
One of the more perplexing problems facing managers is that of deciding on the
allocation of resources to a project that requires a sequence of repeated investments before any or all of the benefits arise. Once some resources have been
allocated to the project, further information
may become available, and the
manager may find out that he is better off by NOT investing further in the project.
But there is a dilemma: by letting go of the project, it will be difficult for the
manager to escape the feeling that the resources previously committed have been
‘wasted’ - even though, before the additional information became available, it
may have been perfectly rational for him to invest in the project’s early stages.
This problem, and the associated tendency to ‘throw good money after bad’, is
commonly known as a sunk cost problem, where the manager is torn between
‘cutting his losses’ part-way through the project or persisting with it in the hope
that, against the odds or the better judgment of a financial analysis, additional
investment will bring the project to a successful fruition.
Economic analysis provides a clear-cut recommendation
to the manager facing
the sunk cost problem. If the objective is to maximize profits, then the allocation
of funds at any point in time should be based exclusively on future costs and
benefits. The amounts invested in the past are sunk costs; neither they nor their
amortization are relevant to today’s decisions, Those who violate this rule are said
to be ‘throwing good money after bad’, leading to the popular label of the sunk
cast fallacy.
It is hard to dispute this line of reasoning. Yet, the apparent fallacious behavior
of throwing good money after bad persists, which justifies its study in economic
settings.
Empirical evidence from the stock market’s response to divisional terminations
(Statman and Sepe, 1989; Parayre, 1993) gives strong indications of the presence
of a sunk cost effect. The market rewards firms with an average positive
abnormal return of 3% to 4% when they finally ‘pull the plug’ on divisions known
to have been struggling in the past, supporting the proposition that managers
persist with projects longer than the economics of the project would dictate. For
example, after accumulating
losses for years on its L-101 1 commercial aircraft,
Lockheed announced in December 1981 that it would finally kill the project.
Lockheed stock jumped 18% the following day. Similarly, following sustained
losses, Texas Instruments announced in October 1983 that it would quit the home
computer business. The company’s stock price jumped 22% the next day.
R. Parayre/ J. of Economic Behavior & Org. 28 (1995) 417-442
419
If the stock market accounts for management’s sunk cost bias, we can also
expect competitors to adjust their strategies in light of the knowledge of an
opponent’s escalating commitment to a losing project. Can one player exploit an
opponent’s sunk cost effect to gain a strategic advantage and improve his position?
Conversely, is it always strategically undesirable, in economic terms, for a player
to display a sunk cost effect? In short, when is (or isn’t) the ‘sunk cost fallacy’
strategically destructive?
This paper focuses on such strategic implications of the sunk cost phenomenon, by formalizing these intuitions in illustrative models of duopolistic
competition. We show that:
- in spite of different behavioral assumptions, formal micro-economic and
game theory models, based on utility-maximization, can be used to derive the
implications of sunk cost behavior;
- even though escalating commitment may weaken the escalating firm, a
psychological commitment to sunk costs can under some conditions be to one’s
advantage.
The next section presents a taxonomy of causes for the sunk cost phenomenon.
This taxonomy presents a number of psychological causes of the sunk cost effect
that complement conventional rationality arguments proposed in the economic
literature. In section 3, we develop a model of a two-person strategic game, and
examine how the presence of the sunk cost phenomenon affects the outcomes in
competitive situations. Section 4 presents conclusions and discusses future research directions.
2. A taxonomy of causes of escalating commitment
Sunk cost behavior may be modeled as rational, even though it sometimes
violates strict economic optimality, by incorporating additional factors (economic
or non-economic) into a utility maximization framework. Modeling behavior in
this way was done as early as thirty years ago by Williamson (1963) in Cyert and
March’s Behavioral Theory of the Firm.
Consider the following examples. First, escalating commitment to a project can
be optimal for a firm as a result of moving down an experience curve, because of
synergies in the firm’s portfolio, if switching costs are high or if sunk capital costs
make a marginal additional investment worthwhile. Escalating the commitment is
justified on purely economic grounds, based on marginal costs and benefits, thus
maximizing the expected net present value of the firm. ’
’ The sunk cost phenomenon, as we have defined it, entails psychological
exceed purely economic switching costs.
costs of switching that
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R. Parayre/ J. of Economic Behavior & Org. 28 (1995) 417-442
The sunk cost effect can also result from inappropriate managerial incentives,
when the manager’s self-interest 3 clashes with the firm’s economic objectives as
they apply to the committed project. As pointed out by Staw and Ross (1987a),
perseverance in the face of adversity is a quality highly valued by society, and
may contribute positively to the manager’s reputation as a decision maker especially under asymmetric information, where the desirability of abandoning a
project is information privately held by the manager. Under these conditions, the
manager’s reputation (which influences his earning potential) becomes an important component of his decisions. A utility-maximizing
manager may well display
the sunk cost effect and persist with a committed project beyond the point that is
financially optimal for the firm. From the firm’s position, a sunk cost effect
results. 4 Kanodia et al. (1989) examine the sunk cost phenomenon in this light, as
an effect driven by information asymmetries between principals and agents that
includes reputation building on the part of the manager. 5
As a further example, consider the dilemma you face on a Sunday afternoon:
you need to work on a paper, but would like to watch the first quarter of a football
game on television. Your prior preferences are ordered as follows:
watching one quarter 5 watching no football $ watching the whole game.
In accordance with these preferences, you settle down to watch one quarter of
football. The problem you face is that after watching the first quarter, your
preferences change so that you want to watch ‘just a few more minutes’ of the
game. After those few minutes, the process repeats itself, and you eventually
become trapped. You end up watching the whole game instead of only one
quarter, and experience regret for having wasted away the afternoon. The change
in preferences may be the result of ‘addiction’ or habit formation rather than of
justifying past investments of time. But just as in a sunk cost situation where a
manager wishes to ‘recoup’ sunk investments, a (momentary) preference change
takes place which influences the decision to persist with a committed course of
action. The topic of preference change and the role of commitment in stabilizing
one’s behavior have been discussed by several authors, and compiled in Loewenstein and Elster (1992).
’ Differing incentives are not new to economic theory: they are at the heart of the principal-agent
problem. Here, reputation effects enter the manager’s utility function as they influence the net present
value of his compensation. These effects may include Iong-term reputation as well as short-term
compensation.
4 Of course, the firm may never know the (ex ante) economic merit of the decision to escalate.
Because of information asymmetries, it can only observe the increased investment in the committed
project.
5 Staw and Ross (1987a) (Staw and Ross, 1987b) discuss some of the organizational solutions,
including appropriate managerial incentives, that can help mitigate the occurrence of escalation.
However, optimal managerial incentives to reduce escalation behavior have not been formally
established. This remains a potentially rich problem for agency theory research.
R. Parayre / .I. of Economic Behavior & Org. 28 Cl995) 417-442
421
Finally, choices may also result from ‘wrong’ or fallacious subjective parameters, distorted by the sunk cost. Substantial empirical evidence supports such cases
of wishful thinking or the illusion of control (Knox and Inkster, 1968; Arkes and
Blumer, 1985).
These examples, presented in decreasing order of economic rationality, offer
some very interesting strategic implications.
3. The sunk cost phenomenon
in duopolistic competition
3.1. A generic model
Firms must often build up capacity before entering an industry. After doing so,
overcapacity may result because of a subsequent shock in the environment which
reduces total industry demand, because of additional entrants, or because of
over-optimistic projections about capacity requirements. An ‘escalating’ firm will
want to use some of its overcapacity despite the fact that it should leave it idle. In
general, the more expensive the capacity, the more a lirrn will want to use it.
Sample industries of recent overcapacity include banks, DRAM chip manufacturing, many chemicals, air transport, as well as firms with high fixed costs such as
extractives (aluminum, steel or soda ash). The wish to recover fixed costs that are
sunk will often lead to the sunk cost ‘fallacy’.
Consider therefore a situation involving two firms (or players), competing as
duopolists in a product market. There are two time periods. At time t,, Player A
must decide on the amount s to be invested (sunk) in a project, into (say) plant or
R&D. At time t,, Player A must determine his play a, simultaneously with Player
B determining her play b. 6 The plays can be investment allocations, production
capacities, quantities produced, or prices. This joint play produces revenues for
each player received at the end of t,, resulting in profits rrA(s,a,b) and 7rB(s,a,b).
Each player is seeking to maximize his/her own utility function. The key
departure from a traditional neo-classical economic analysis of the problem lies in
the arguments that may enter Player A’s utility function. In an analysis of the sunk
cost phenomenon, considerations other than just the impact of marginal profits (or
future profits, exclusive of the sunk cost) must be part of Player A’s utility
function. Since Player A’s commitment to the project stems from the cost s that he
has sunk into it, the sunk cost s must therefore become an explicit argument in
Player A’s utility function, along with marginal profits.
6 This analysis extends to other situations where the two players must allocate funds to a project
common to both players, such as a joint venture or a competitive project. In this case, the players’
decision at t, is to determine how many additional funds to allocate to the project. Moreover, as each
player may also have alternative investment opportunities, the profits from this common project am net
of all the other activities each player may be involved in.
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R. Parayre / .I. of Economic Behavior & Org. 28 (1995) 417-442
In this two-period situation, we can think of s as determining which of several
Nash games will be played at t,. Player B has no control over the amount s sunk
by Player A at t,. Because of the recursive determination of A’s optimal s * from
the Nash solution at t,, Player B can only indirectly influence A’s choice of s
through her allocation decision at t,, using her knowledge of A’s failure to ignore
sunk costs in determining that allocation. Because B cannot precommit or signal
any departure from her Nash allocation prior to her decision at t 1, she will have no
choice but to play her Nash move in the game chosen by A through A’s choice of
s *. Player A therefore enjoys a clear first-mover advantage, even though final
outcomes ultimately depend on the utility functions of both players.
We can examine the situation at t,, in which Player A has already sunk s at t,.
The optimization problem at t, is:
for Player A: max u*(a, b *, SX # n*(a, b *, s) if the sunk cost effect applies)
B
for Player B: max 7rB(a*, b, s),
b
where a [and b] represents Player A’s [and B’s] play at t,.
With s being Player A’s sunk investment, and if a is quantity produced (or
production capacity), escalation of commitment can be expressed as u,“,> 0. That
is, as the sunk cost s in a project increases, the marginal value of escalating
commitment in that project also increases. ’ This commitment, which may stem
from behavioral considerations discussed in section 2, enters the manager’s utility
function at time t,. 8
A comparative statics analysis of the Nash game played at t, provides insight
into the impact of Player A’s sunk cost effect on the resulting equilibrium
allocations, and on the strategic advantages or disadvantages that emerge. Situations where the decision variables are quantities produced, production capacity, or
further investments in a project, usually exhibit downward sloping reaction
functions. Taking first-order conditions, differentiating them totally with respect to
s and using information from the second-order conditions 9, for downward sloping
reaction functions we get:
da*
db’
ds > 0 and ds <o.
: For our purposes, this need only hold locally, around the equilibrium.
In many cases, this characterization may be based on purely economic considerations - for
example, under significant inteltemporal economies of scale in project #l. Under intertemporal
economies, one is better off investing smaller amounts sequentially in the project, either because
learning occurs or because congestion diseconomies make a huger one-time investment in the project at
t, uneconomical. Similarly, sinking s at t, may reduce A’s marginal cost of production at t,.
’ Second-order conditions require that at the Nash equilibrium, u& < 0, atb < 0 and I&T& u$qi > 0. In addition, downward sloping reaction functions require u& naF < 0.
R. Parayre / J. of Economic Behavior & Org. 28 (1995) 417-442
423
Thus as Player A’s sunk costs in project 1 increase, A’s tendency to persist
with project 1 also increases - accompanied by a shift in his reaction function. B’s
profit function, on the other hand, and hence her reaction function, are unaffected
by a change in s (except through the induced change in a’). An increase in a*
therefore leads to a decrease in b’ as we move southeast along B’s downward
sloping reaction function. When the direction of increasing profits is towards the
monopoly solution, as in most Cournot duopolies (where firms compete on
quantities), then the following holds at the Nash equilibrium: As A’s sunk cost s
increases, 7~A (excluding s) increases, while r* decreases. Hence, it is not to B’s
advantage for A to increase his sunk cost commitment s, as A’s escalation effect
makes Player B worse off. This result lies at the core of our signalling application,
which we now develop.
3.2.
Sinking
costs as a precommitment strategy: using your sunk cost effect to your
advantage
Spence (19771, Spence (19791, Dixit (1979) and Dixit (1980) demonstrated the
economic benefits of sinking costs into idle production capacity as a means of
deterring entry. Ghemawat (1991) also stresses the broader economic importance
of commitment as a strategic tool. While economic deterrence is quite compelling,
we argue that there is also economic value in sinking costs as a psychological
deterrent. We therefore model the implications of sunk cost behavior resulting
from psychological causes - as opposed to earlier economic explanations or to
reputation concerns in repeated games played by profit-maximizing players. lo We
examine the psychological importance of commitment in strategic situations,
which in turn will influence the economic outcomes of the firm making the
commitment.
We illustrate this via a simple Cournot duopoly signalling example I’, which
builds on the intuition from the preceding comparative statics. Consider the
following situation. Player A has sunk s into the development of a product which,
at t ,, he will sell in a duopoly with Player B. Moreover, Player A can be of one of
two types:
Type N: Displays 120sunk cost effect
Type S: Displays a sunk cost effect.
Type being information private to Player A, Player B starts off by assuming a
probability distribution about A’s type, based on a population base-rate or on
Player A’s reputation for having escalated commitment in the past. B then
‘” We do however preserve ‘internal consistency’ - where at any point in time, decisions are those
which maximize. one’s utility function at that time. We also preserve the recursive logic of dynamic
programming.
” For the fundamentals on incomplete-information games of this type, see Harsanyi (1967-68).
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R. Parayre / J. of Economic Behavior & Org. 28 (1995) 417-442
observes the amount s that A sinks into the development of the product, and with
this information updates her beliefs about A’s type. Using this updated (posterior)
distribution, B then goes on to play a simultaneous Nash (Cournot) output game
with A.
The price of the (jointly produced) product is:
p=-y-6(a+b),
(y,E>O).
With little loss of generality, costless production is assumed (for simplicity), as
is no time discounting between t, and t, (as would be the case for very short time
periods).
We can solve for the equilibrium solution(s) of this game recursively, in three
distinct steps. First, we solve B’s output game. Let p be p(s), B’s posterior about
A’s type ( p also known to A). Then B’s output decision is to
My[y
- 8(a + b)]b
s.t.
A plays
argmax[ y - S(a + b)] a
and plays
argkax[r
a
(a(s)
da/ds>O)
20;
with probability ( 1 - p)
- S(a + b)]a + a(
with probability p.
That is, B maximizes rrB under the condition that A of type N maximizes rr A
and A of type S maximizes uA = +r~’+ o(s)a. The term a(s>a is a particular
parametrization of escalation of commitment - where an increase in s encourages
increasing a, thus leading to u:~ > 0 in this Coumot case. This is our fundamental
characterization of a ‘sunk cost effect’, and is intended to capture the increased
psychological commitment associated with an increase in s.
B’s optimal production is found by solving the first-order condition for B
simultaneously with the first-order condition for A. Given the uncertainty about
Player A’s type, Player B must take Player A’s (uncertain) production to be
a* = (1 - p)ah + pa: I
This is shown in Fig. 1.
This graph shows the optimal reaction functions for Player A of Type N, of
Type S, and for Player B, and the resulting optimal production for B (point P *>
given the uncertainty about A’s type. Point P * is located at (1 - p)P, + pPs. We
obtain
b* = Y - pa(s)
36
.
R. Parayre / J. of Economic Behavior & Org. 28 (1995) 417-442
425
b*=
a
Fig. 1. Player B’s output decision via reaction functions.
Player A’s production will then be a function of his own type. Solving A’s
first-order condition simultaneously with b’, we find that A of type N will
produce
* _
aN -
2Y + Pa
66
with
u;=nN--
TB=
* _ 4Y2 + 4PW(S)
+ p2a2(s)
366
2y* - pya!(s) - pW(s)
186
Similarly, A of type S will produce
* _ 2Y + (3 + p)a(s)
as 66
R. Parayre / J. of Economic Behavior & Org. 28 (1995) 417-442
426
yielding
* _ 4y2 + (12 + 4p)ya(s) + (p + 3)2aZ(s)
us 363
na _- 2Y2 - (3 + P)V(S) + (3P - p2)a2(s)
188
The interesting result at t, concerns r*, the marginal profit part of u*:
* _ 4Y2 + 4PYQ)
=s -
+ P2a2w
366
-- ~2(s)
46
.
In this case, any a(s) < 4py/(9 - p2> will increase rt. I2 Hence, Player A can
actually be belter ofSfinan&lly at t, than in the case where A is known to be
maximizing rr * only, even though he displays the sunk cost effect - a positive
by-product of B’s uncertainty about A’s type. This occurs because B reduces her
Nash production to account for the possible sunk cost effect of A.
This result addresses the impact of the sunk cost effect as of t,, once s has
already been committed. But should Player A deliberately sink s at t, to begin
with (assuming it is not necessary), in order to then use his sunk cost effect at t, to
his competitive advantage? Can Player A be made better off by such a strategy
even at t,? We can find out by comparing the profit function at t, with and without
a sunk investment s. With no discounting between periods, Player A’s profit
function at t,, when investing s, equals: I3
4y2 + 4pyc!L(s) + p%?(s)
ForAofTypeN:
n$=
ForAofTypeS:
7~:=
368
472 + 4pyo!(s) + p%?(s)
366
-S
02(s)
-4s-s
The optimal amount s* depends upon the specific values of the problem
parameters. Characterizing this equilibrium requires that we consider 3 possible
parameter scenarios. These are illustrated in Fig. 2.
Case I
TN” and rt
are both maximized at .sA= si = 0. This is the equilibrium.
Case 2
rri is maximized at sl; > 0 and nt is maximized at si = 0.
Then any s > 0 signals with probability 1 that A is of Type N, so that p = 0.
Hence both types of A are best off playing s * = 0. This is again the equilibrium.
I2 Non-negativity conditions for b constrain the allowable parameter range to be y 2 pa(s).
I3 Note that for any positive s and the associated positive (I(S), ?T: > w[. When s = 0 and hence
(Y(s) = 0, ?r; = lrs” = y*/9s.
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R. Parayre / J. of Economic Behavior & Org. 28 (1995) 417-442
427
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R. Parayre /J.
of Economic
Behauiar & Org. 28 CI995) 417-442
Case 3
rr: is maximized
at sl > 0 (and because ri > ,rr.$‘Vs, rr$ is maximized
sk > 0). This is the richer, more interesting case. It occurs when
4PW(S)
+ p202(s)
368
02(s)
- 46
at
-s>O(forTypeS)
(and hence, automatically
4PYdS)
+ p2a2(s)
366
-s>O(forTypeN)).
Here, both A Types N and S have an incentive to increase p, which yields
greater profits. Hence, it is to A’s economic advantage (in terms of purely
economic profits), regardless of his type, to put himself in a sunk cost situation at
t, - a first-mover advantage. A then benefits from an increased a* in equilibrium,
under the threat that he may be maximizing a utility function at t, that includes a
sunk cost effect component. This advantage to a player A of type S thus stems
from the output adjustment it triggers in B; an aa’ditionaf advantage to a player A
of type N stems from B’s belief that A may be of type S.
Intentionally
sinking (and losing) costs thus becomes beneficial because it
convinces an opponent that you may not be a perfectly rational adversary (from a
narrow rA perspective) in the next time period - which is something you may be
able to exploit. The following results characterize the equilibrium of this signalling
game:
Non-existence of a separating equilibrium
Because 7~; * > rt * > y2/9S (the latter holding when s = 0), there exists no
s > 0 that would make Type S better off than s = 0 while making Type N worse
off than s = 0 (see Fig. 2, Case 3). Therefore, a Type S can never shake himself
free of a Type N trying to mimic him - which a Type N will always want to do
(to increase p>, as any sN # sg would signal that A is of Type N and would push
p to 0.
Uniqueness of the pooling equilibrium
Because si maximizes n=c given p, it also maximizes p (it signals that the
amount s being played is the best move for a player of Type S). From the
preceding paragraph, si is also the best move of a Type N. Hence, the unique
optimal s for both types of A is si. This is a non-informative allocation, so that
Player B’s posterior about Player A’s type = p = Player B’s prior about Player A’s
type. Hence, p is an exogenous parameter which, in conjunction
with the
particular values of y. S and cr(s), will determine which Case (1, 2, or 3) is being
played.
R. Parayre / J. of Economic Behavior & Org. 28 (1995) 417-442
429
A’s sunk cost, regardless of A’s type, will be
and
= 0
= si
in Cases 1 and 2,
in Case 3.
The subsequent equilibrium output allocations of A and B will then be made, as
described above. Note that a nonzero s * invested (in Case 3) by Player A of either
type makes him better off, while making Player B worse off than if s = 0.
The sunk cost effect thus provides Player A with an interesting means by which
to gain a strategic edge. By sinking money into the project at t,, Player A of Type
S precommits himself to displaying a sunk cost effect in the ensuing game, and in
effect, to behaving ‘irrationally’ (from a pure profit perspective) at t, - in essence,
strategically manipulating his own preferences, in order to be better off financially
in the end. To quote Schelling (1960):
‘It may be perfectly rational to wish oneself not altogether rational ’ (p. 18).
While Schelling made that statement in the context of making threats credible, it
also applies to our case of sunk costs. The model just developed formalizes that
intuition. By sinking costs at t,, Player A signals the possibility that he will
maximize uA rather than g A at t,. If Player A can successfully convince Player B
that he is subject to a sunk cost effect and play an apparently - from Player B’s
perspective - irrational strategy, then A can gain a strategic advantage. To do so,
A must precommit himself to sinking s into the product at t, and must convince B
that the effect of this sunk cost will be (Y(S)in his utility function. Of course, in
order to convince B it may be necessary for A (of Type S) to actually fall prey to
a sunk cost effect - a strategy in the spirit of true precommitment. In that case, no
misrepresentation occurs: the advantage to A results because of, and in spite of,
his own sunk cost effect. Frank (1988) presents a compelling, comprehensive
discussion of the adaptive value of what appear to be ‘irrational’ choices, and also
addresses the problem of mimicry (e.g., a Type N trying to pass off as Type S).
This subtle finding of a ‘rational irrationality’ requires further elaboration. In a
two-period model, the problem of bridge-burning, as an example, assumes a
constancy of purpose over both periods. Indeed, you burn your bridges at t, to
ensure that your choice at t, maximizes the utility function of t,. Similarly,
Ulysses asked to be bound to the mast of his ship to ensure that his ‘irrational’ self
at t, did not undo what his ‘rational’ self wanted at t,. By constraining the choice
set available at t, , a strategic precommitment thus ensures consistency of actions
with the preferences of t,. Subgame perfection is thus imposed by restricting
flexibility at t, to choices which are optimal from the ‘rational’ perspective of t,.
In contrast, our sunk cost player is allowed, depending on his type, to display
his ‘irrationality’ (of maximizing u instead of rr) at t,, rather than constraining
that possibility away. B must then adjust her output to account for that possibility.
Given that adjustment, A would always be better off maximizing 7~ at that point.
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R. Parayre / J. of Economic Behavior & Org. 28 f 1995) 417-442
But under this strictly ‘rational’ subgame perfect solution, B would never adjust
her output. The advantage to A occurs because A cannot consciously choose his
type at t,. The possibility of being ‘irrational’ (i.e., sub-optimal) at t, is what
opens the door (in some cases) to the possibility of greater profits for A of either
type.
Thus to maximize rr over two periods, you may need to force yourself, or leave
yourself vulnerable to, maximizing
u (# n-) in the second period. This result
differs from earlier rational explanations or implications of sunk costs. Sunk costs
as precommitment
thus take on a very different, more behavioral, connotation
here.
It is legitimate to ask whether a manager’s sunk cost effect is a credible
commitment for firm A to be of Type S. Why, for example, shouldn’t firm A fire
a Type S manager once the deterrence effect is known to B? Two possible answers
come to mind. If firm A establishes a reputation for firing Type S managers before
t, (and always playing aA at t,), it will quickly lose the benefit of this deterrence
effect in future similar encounters against other competitors (in a multiple-encounter world). Alternatively,
as demonstrated in Kanodia et al. (1989), escalation of
commitment can arise out of information asymmetries and agency concerns on the
part of the manager. Firm A shareholders may therefore be unable to know the
manager’s type before t, any better than the probabilistic knowledge which firm B
possesses in a single-encounter
world. In either case, sinking costs can constitute a
credible commitment to Type S behavior in the output game at t,.
Arkes and Blumer (1985) recognized that the sunk cost effect can be used to
one’s advantage. They quote an argument made by Dowie (1981) in the context of
nuclear energy. He suggested that if construction of power plants can be secretly
initiated, sunk cost arguments can then be brought up when the public finds out, to
argue that it is too late for construction to be stopped. While somewhat cynical,
this example does point out the potential for Player A to impose a sunk cost either to himself or to Player B - as a strategic move that permits him to get his
way. Self-imposing sunk costs demonstrates resolve for persisting with the project.
Imposing sunk costs on someone else, which would be the case if public funds
were used in construction of the power plants, transfers the sunk cost effect to
others. l4 This recognizes sunk costs as an important strategic tool in delegation or
principal/agent
situations. This is an important problem, with significant potential
for analysis.
A more general formulation of sunk costs as precommitment
is represented
generically as a game in extensive form, in the game tree in Fig. 3.
Player A must first decide how much of a cost s, if any, to sink into a project.
Some uncertainty
then gets resolved: If successful, Player A can then reap
I4 Prior investments may in fact make persisting
alternative at that point.
with construction
of the plants the most desirable
R. Parayre / .I. of Economic Behmior
monopoly
profits
4”
& Org. 28 (1995) 417-442
431
U-Cl
A
A
A maximizes
AA
(u*, nBla;,b*)
(x*. xBla;,ti)
Fig. 3. Detailed game tree for sunk costs as precommianent.
monopoly profits - having pre-empted Player B’s involvement in the project. If
unsuccessful, Player A loses the initial allocation of s and then enters a simultaneous Nash game with Player B. It is the unsuccessful resolution of the uncertainty
which drives the psychological motivations leading to a sunk cost effect for Player
A - and the resulting change in his utility function. Player A, which may not have
perfectly anticipated his reaction to sunk costs (hence his prior q* ), ends up in one
of two utility states (i.e., with one of two utility functions) following these sunk
costs. Player B does not know the utility state of Player A, as this is A’s privately
held information. She only has a prior probability estimate, qB. A and B then
proceed to play the Nash game of the second period along the lines indicated in
the bottom half of the figure, yielding the equilibrium allocations. The payoffs at
the bottom of the figure reflect A’s different utility functions. l5
432
R. Parayre / J. of Economic Behavior & Org. 28 (1995) 417-442
3.3. Price competition
In the case of Bertrand (price) competition, usually associated with heterogeneous products, having sunk s (into plant, say) can motivate the manager to try
and operate as close to full capacity as possible (so as not to admit that the plant
may be oversized). So while the decision at t, is one of pricing, the sunk cost
effect will reward increasing production, and hence the manager wants to decrease price. So when the decision variables a and b are prices rather than
quantities, escalating commitment can therefore be expressed as u$ < 0. I6
Price competition generally produces upward sloping reaction functions. For
upward sloping reaction functions resulting from price competition, we get “:
db’
da’
ds < 0 and ds <o.
A decrease in a* thus leads to a decrease in b ’ as we move down B’s upward
sloping reaction function. ‘*
Determining which player ends up better off because of A’s sunk cost effect, or
as the sunk amount s increases, depends on the direction of increasing profits
along the players’ reaction functions, which in turn depends on the elasticity of the
demand curves. This requires additional structure to be imposed on the problem.
Consider for example the following simple parametrization of a Bertrand
duopoly, which has each firm’s demand linear in both firms’ prices:
qa = y - 6p, + a(p, - p,)
demand for As product
qb = y - 6p, + o(p, - pb)
demand for B’s product
where pa, p,, are the prices for A and B’s products.
In this case, Player A will r9
max uA = 7~~ + a(s)qa = q,p, + o(s)q,
Pa
= [Y - SP, + o(p, - P,)](P, + o(s))
(4s)
2 O;do/ds ’ 0)
*’ A similar albeit more extensive version of this tree could be used to describe. escalation of
commitment in the context of the ‘dollar auction’ game.
I6 The case of price competition may be less compelling and possibly less prone to sunk cost
behavior, as a pricing decision is not literally one of escalating commitment.
” Upward sloping reaction functions require u$ , ?r,f:> 0. The results are then derived in a manner
similar to that described in section 3.1.
‘s Milgrom and Roberts (1990) suggested different economic examples of upward-sloping reaction
functions. R&D races, oil drilling and even arms races display upward-sloping reaction functions. In
these cases, where the decision variables are.quantities, the sunk cost effect is expressed as u$ > 0, and
the comparative statics cannot be. signed.
I9 Here. cr(s)q, leads to u$~< 0, capturing escalation of commitment in this Bertrand case.
R. Parayre/ J. of EconomicBehavior& Org. 28 (1995) 417-442
433
and Player B will
maxnB =
qbpb
=
[Y
-
‘Pb
+
+a
Pb)]Pb-
-
Pb
Taking first-order conditions and solving simultaneously, the Nash equilibrium
prices are computed to be
-2a.(s)s*
+ 2&y - 4a(s)6a
P: =
+ 3ya - 2a(s)a2
46* + 86a + 3u2
pb’ = -
-26y+a(s)6u-3yu+a(s)u*
(26 + a)(2S + 30)
yielding
uA_ (8+u)]2u(s)6*+26y+4a(s)8u+3yu+a(s)u*]*
(26 + a)*(26 + 3u)*
7FA= -
(6 +a)
(26 + a)*(26
+ 3u)*
. [j’ - k* + 3ja(s) u* + ka(s)u*
+ 2a2(s)u4]
wherej=(2S2+4&r)a(s)>Oandk=y(26+3g)>0;
yrB _ (6 + u)[ -280 + a(s)& - 3yu + a(s)u’]’
(26 + a)*(26 + 3u)*
While u* is increasing in a(s), we find both yr* and 7rB to be decreasing in
a(s). Therefore, in this simple Bertrand case, both A and B lower their prices and
both end up worse off because of A’s sunk cost effect. If s has already been sunk
(in anticipation of greater demand than actually obtained), A’s sunk cost effect
makes him worse off. If s has not already been sunk, it is not to A’s advantage to
intentionally sink s in order to signal that he may be of Type S.
3.4. Exploiting an opponent’s sunk cost effect 2o
We now turn our attention to the problem of exploiting an opponent’s sunk
costs. Under what conditions will one player be better off because of an opponent’s
2oThere is an ongoing debate concerning the (lack 00 dynamic consistency of non-expected utility
agents, and how they can be exploited in economic decisions under uncertainty.For a thoroughreview
of this debate, see Machina (1989). While not concerned with the same type of behavior, our present
context of changing preferences following sunk costs raises some similar philosophical issues.
434
R. Parayre/
J. of Economic Behavior & Org. 28 (1995) 417-442
sunk cost effect? The impact of one player’s sunk cost effect on an opponent in
general will depend on the particular economic problem being studied.
It is natural that under the conventional single-product Coumot assumptions of
section 3.1, B cannot profit from A’s increased commitment to the product. But by
extending the situation to a Cournot duopoly with rwo distinct products at t , , there
now is the possibility for B to exploit A’s increased commitment in product #I
(and away from product #2) by focusing more of her investment on product
#2. 21 The presence of a second product in which the duopolists also compete
thus provides an opportunity for B to benefit from A’s sunk cost effect.
We can work out an example of such multipoint Cournot competition. As
before, Player A has sunk s into the development of product # 1 which, at t ), he
will sell in a duopoly along with Player B. But there is also another product,
whose demand is independent of the first, which A and B also sell in a duopoly.
At t,, Player A has an amount A to invest and Player B has B. 22 Player A puts a
into product #I and A - a into product #2; Player B puts b into product #I and
B - b into product #2. The price of product # 1 is: p = y - S(a + b), (y, S > 0).
The price of product #2 is: q = 0 - a[(A - a) + (B - b)], (0,~ > 0). Once again,
no costs of production are assumed, so that the optimization problem is:
Player
A: max[y - 6(a + b>]a + [f3 - a(A + B - a - b)l(A - a) + o(s>a,
B
((Y(s) 2 0)
Player
B:max[y - &(a + b)]b + [fl - a(A + B - a - b)l(B - b).
b
Derivation of the Nash equilibrium allocations and profits is once again
obtained by solving the two players’ first-order conditions simultaneously. The
resulting profits (at t, 1 are:
+9B( 68 + yo - &[A + B])]
*’ The issue of multipoint competition has been studied by Bulow et al. (1985) in their economic
analysis of multimarket oligopolies.
*’ This budget constraint on Player A is necessary to introduce a dependence between the allocations
on the two projects. Without some form of constraint, Player A’s sunk cost effect on product #l would
not affect his profits and optimal allocation on the independent product #2. The results of any sunk
cost effect in this case would be driven purely by competition in market #I, replicating the results just
derived in the single-product case. In the extreme, in the absence of such a budget constraint Player A
could sink money into both projects, and benefit from his own sunk cost effect in both markets! But in
many organizational contexts, it is quite reasonable to talk of a budget constraint, as divisions are often
allocated fixed pots of money and divisional managers ate usually limited to those fixed amounts especially when the time periods (and the associated windows of opportunity for investment) am short,
and managers have no time to raise additional funds.
R. Parayre/ J. of EconomicBehavior & Org. 28 (1995) 417-442
435
a(s)<0 not allowed
The shaded areas constitute the feasible
regions for producing non-negative
quantities.
-3aA -
There are 3 possible regions:
Region 1 : n*? in a(s) and T? 1 in a(s)
Region
II : 24 in a(s) and n?J in a(s)
Region III : K*J. in a(s) and # tin a(s)
Fig. 4. Regions of exploitability for the deterministic two-product example.
n*=
g(s~u)[-2ui(s)cu(s)(Y-8)+(y-~)2
+9A(60
-I- ya - 6a[A
+ B])]
Fig. 4 shows the different regions that make each player better off or worse off
at t, because of player A’s sunk cost effect. In this deterministic 2-product case in
which player A is known to both parties to fall prey to a sunk cost effect, the
benefits of the effect can accrue either to A or to B (but never to both), or to
neither player. 23
For example, the case in which y= 8, i.e., when the demand curves for both
products have the same intercept, we are in region III of Fig. 4. Player B’s profit
R. Parayre / J. of Economic Behauior & Org. 28 (1995) 417-442
436
increases in CX(S)while A’s decreases, so that B is made better off by Player A’s
sunk cost effect. In this situation, Player A suffers from a first-mover disadvantage, which Player B can then exploit.
So, unlike in our previous examples, we cannot draw unequivocal conclusions
about the (dis-ladvantages of the sunk cost effect. Yet even in this simplest of
two-product situations, with costless production and linear demand, the sunk cost
effect may offer potential for exploitation by one of the two parties.
Extending this game to a signalling one yields much the same conclusion as our
single-product signalling example, derived in much the same way. Profits at t, in
the two-product signalling game are:
~2(4
A==A_
TS
N
4(S+a)
A will now sink s * > 0 only when the parameters are in the region which
makes both types N and S better off (Case 3 in section 3.21, and under those
conditions will sink s = sS . For any other parameter set, A will sink s * = 0, and
no sunk cost effect will take place.
For example, if both products have identical demand curves (i.e., y = 0 and
6 = v ), the profits at t, are given by:
mi=yA-G(A+B);
+
p202 (s)
which is increasing in o(s) and in p
726
a,A=yA-G(A+B)$
+
(P’ - 9)02(s)
726
which is decreasing in o(s) t/p.
These equations show A of type S to be made worse off (rr,” is decreasing in
(Y(S)Vcu(s)) because of his sunk cost effect - regardless of B’s belief p about A’s
Type. Therefore, while A of Type N benefits financially from p > 0 and (Y(S)> 0,
A of type S never has an incentive to sink any s at t, (or to ‘bum money’, as it
were 24>,thus making the equilibrium fully revealing: A of type S always sinks
s = 0, and the normal economic Nash equilibrium holds at t,. Any s > 0 tells B
that A is a Type N hoping to pass off as Type S (i.e., trying to misrepresent his
type), which has for effect to push p to 0, again resulting in the economic Nash
equilibrium at t,. Hence, A (regardless of type) can never be made better off by
his sunk costs. Any s > 0 (including s sunk inadvertently) will be lost, and any A
of type S will only be made worse off financially by his sunk cost effect.
23
This graph is based on S > (r and A > B. The three regions look slightly different but the
conclusions remain unchanged when 6 < u and/or A < B.
24 Van Damme (1989) addresses the strategic role of ‘burning money’ as a means of selecting an
equilibrium via forward induction - a problem quite different from the one addressed here.
R. Parayre / .I. of Economic Behavior& Org. 28 (1995) 417-442
437
3.5. Welfare implications
If A and B were playing a cooperative game, the first-best outcome for them
would obviously be to operate jointly as a monopoly with some sort of sharing
rule. But this solution is ruled out in a non-cooperative environment, where the
duopoly solution prevails. Yet given the possibility for one player to be made
better off by a sunk cost effect, we can ask whether joint welfare can be improved
if one party has a sunk cost predilection.
In the single-product signalling game in section 3.2, joint welfare can indeed be
greater than in the purely ‘rational’ duopoly setting. When B adjusts her output to
account for the possibility of A’s sunk cost effect and A turns out being of the
‘economic’ Type N, joint profits r* + 7~~ are 2y2/96 + p&)(27 p&1)/36&
which exceed the total duopoly profits under no sunk cost effect of
2 y2/96. 25 If A is the ‘escalating’ Type S, joint profits are 2y2/96 + [(2p 6)ya(s) + (6~ - p2 - 9>(r 2(s>]/36S, which is always less than in the purely
rational duopoly case.
A similar conclusion arises in the situation in which both A and B can display a
sunk cost effect. In this case, where both players sink costs in the first time period
and Player B also has some chance of being of the ‘escalating’ type, there is an
economic pressure for each player to decrease output (to adjust for a possible
escalating opponent), and a psychological pressure to increase output if one is of
the escalating type. This situation now contains four different reaction functions:
for A,, As, B, and Bs, with the resulting equilibria (ai,bG), (ai,b,‘), <ai,bi),
(ai ,bG 1. See Fig. 5. Analytical results can be derived in a manner which closely
parallels section 3.2.
Assume that both players share an equal probability p of being of Type S and
to display a sunk cost effect of magnitude (Y(S).Under these conditions, (ah,bA) is
located southwest of the purely economic duopoly solution located at the intersection of A, and B, (point e in Fig. 51, and has greater joint welfare than point e if
pa(s) < y/2 (i.e., if there is a small chance of a player being Type S and/or the
sunk cost effect is weak). Otherwise joint welfare will be less than at point e. Note
that A of Type N now has to decrease output to account for the possibility of B
being of Type S. Thus the threat of a Type S opponent in effect disciplines players
into reducing their output levels. Under the right conditions, 26 this brings them
closer to the first best solution of joint monopoly output, making them both better
off than if they both knew each other to be ‘perfectly’ rational! The other three
equilibrium allocations (ai ,bl 1, (ai ,bi > and (al ,bi > can be shown to result in
” The development in section 3.2 required that y > PLY(S)as a non-negativity condition, leading to
this general conclusion.
26 If an opponent’s expected sunk cost effect is strong, players may reduce their output levels below
the joint monopoly output, and may actually be worse off than at the economic duopoply level.
438
R. Parayre/ J. of Economic Behavior & Org. 28 (1995) 417-442
a
Fig. 5. Cournot signalling game with two players of unknown type.
welfare losses when compared to point e, as they all entail greater joint
production, and thus produce smaller joint profits.
In the two-product case of section 3.4, joint welfare increases when (Y(S)<
-(ye> -- the re gion to the southwest of the line of slope - 1 bisecting the
bottom-right quadrant in Fig. 4. Anything to the north of that line results in
reduced joint welfare.
joint
3.6. The strategic manipulation of an opponent’s preferences and other types of
commitment situations
In the preceding examples, any advantages to Player B are purely by-products
of A’s sunk cost effect, as B has no control over the amount s sunk by A at t,. 27
But A’s sunk cost effect can present yet another set of opportunities to Player B.
By moving first, B may be able to strategically manipulate A’s preferences to her
own advantage by actively influencing A’s choice of a sunk amount s. This will
allow B to actively encourage and exploit Player A’s sunk cost effect.
Examples of such strategic manipulations abound. Just as an economic (sunk
cost) commitment to a project results in greater psychological commitment to it (in
27
unless B too can mis-represent her true preferences at t, - a situation which we do not address
here.
R. Parayre/ J. of EconomicBehauior& Org. 28 (I 995) 417-442
439
effect, increasing the utility of a positive outcome from that project), psychological commitments will often have the same effect. In essence, past commitments,
be they economic or psychological, often increase future willingness-to-pay. 28
Consuming heroin increases an addict’s need (and utility) for future consumption
of the drug, hence increasing his willingness-to-pay. Car dealers low-ball (Cialdini
et al., 1978) customers by having them agree to a base purchase price on a car,
only to come back with a higher price (or with high-priced options) after
‘checking with the manager’ who would not agree to the initial low price.
Charitable organizations have you commit to giving a small amount, and then
escalate their request, with great success - a strategy known as the foot-in-the-door
technique (Freedman and Fraser, 1966). Stores offer huge discounts on some
products to catalyze high-priced tie-in sales - or loss-leader pricing. Or they
advertise a ‘super-special while quantities last’ to bring customers to the store, and
quickly run out in order to sell the product or a substitute at much higher prices a bait and switch strategy.
All of these situations stress the influence of psychological commitment on
increasing consumers’ willingness-to-pay. In each of these cases, Player B moves
first and tries to have A commit psychologically to a product or idea. If successful,
this strategy can increase A’s willingness-to-pay. Each of these situations can be
modeled as a three-period game, where B moves first, tries to induce commitment
from A, and follows up that commitment with a higher-priced offer. Each situation
offers the potential for true strategic preference manipulation on the part of an
opponent. These games are a topic of ongoing work.
4. Conclusion
This paper has addressed a problem which, despite its economic importance,
has been largely ignored in economics and strategy. The sunk cost phenomenon the tendency to overinvest in previously committed projects - has long been one
of the most persistent problems in decision making. The frequent warning that
‘sunk costs shouldn’t matter’ in determining future allocations, made in just about
every managerial accounting or finance textbook, is an acknowledgment of the
pervasive nature of this phenomenon. Yet there have been few attempts made at
systematically modeling its causes or its implications.
This paper has taken the first steps toward a formal, systematic analysis of the
implications of the sunk cost phenomenon. We have modeled some strategic
effects of the sunk cost phenomenon in order to make behavioral predictions and
provide prescriptive advice. We have demonstrated that even in the simplest of
*s Economic models of addiction have been proposed by Becker and Murphy (1988), and discussed
in Loewenstein aad Elster (1992).
440
R. Parayre / J. of Economic Behavior & Org. 28 (1995) 417-442
competitive settings, such as two-period duopolies, a sunk cost effect could lead to
some interesting and somewhat counter-intuitive
insights. Enrichments
such as
additional dynamics, more uncertainty,
and population ecology considerations
could be introduced. These added degrees of freedom would increase the range of
potential strategic implications of sunk cost behavior. These can be explored in
future research.
Market arguments could be used to contend that the presence of strategic rivals
would impose economic rationality on otherwise escalation-prone
agents - ‘irrational’ firms being driven out of the market, thus constraining sunk cost behavior
to very isolated cases. It is indeed legitimate to ask whether managers engaged in
intensely competitive environments display reduced, the same or greater sunk cost
tendencies as do individuals
facing personal decisions, or laboratory subjects
responding to hypothetical
questions (as in Arkes and Blumer, 1985). Some
evidence (Brockner and Rubin, 1985) suggests that competitive situations actually
aggravate the sunk cost tendency - the concept of ‘winning at all cost’ overriding
economic rationality. When combined with the managerial reputation and self-presentation motives that exist in organizational settings, the potential for sunk cost
behavior among competing firms is great indeed. Teger (1980) has presented
empirical results on the ‘dollar auction’, and has pointed out the importance of
sunk cost considerations in the escalation of warfare.
The sunk cost phenomenon may even have value in single-agent situations. For
example, the knowledge of one’s own escalation tendencies forces one to consider
the initial investment decision very seriously. Prelec and Herrnstein (1991) also
argue that adherence to specific decision ‘principles’ is valuable because of the
inherent difficulty in continually
making cost-benefit calculations. The vow of
marriage, for exampIe, can be viewed as a public declaration of adherence to the
sunk cost fallacy, which eliminates the need for future cost-benefit analyses.
The modeling approach in this paper can extend beyond the sunk cost problem
to analyze other behavioral departures from conventional economic rationality, in
problems located at the interface between psychology and economics. In particular, the issue of preference change is a very rich, yet largely unexplored topic in
economics. (For a notable exception, see Loewenstein and Elster, 1992.) In this
paper, it was assumed that one could perfectly anticipate one’s own future change
in preferences that follows sunk costs, and by so doing could mitigate the negative
implications of the sunk cost effect (or even extract positive ones) by adjusting
one’s sunk costs at t,. Yet evidence from social psychology (Nisbett and Wilson,
1977) suggests that people will likely be poor predictors of their own future
behaviors. What’s more, as discussed by Akerlof (1991) in a paper on procrastination and obedience, many situations involve changes in preferences that were not
foreseen as of t,, resulting in pathological or even destructive outcomes. This is an
important problem, which could be addressed within our present framework.
Fundamental
questions about the pervasiveness
and importance of sunk cost
behavior in competitive settings also remain open. Is an individual’s sunk cost
R. Parayre /J.
of Economic Behavior & Org. 28 (1995) 417-442
441
behavior a generic tendency or is it strongly situation-specific? Do managers, in
practice, exploit (expectations of) sunk cost behavior to increase profits? And what
is the impact of competitors and of the intensity of competition on sunk cost
behavior? These are all important questions which should be subject to empirical
investigation.
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