Peace Economics, Peace Science and
Public Policy
Volume 15, Issue 1
2009
Article 8
A Note on Second Order Probabilities in the
Traditional Deterrence Game
Lisa J. Carlson∗
∗
†
Raymond Dacey†
University of Idaho, [email protected]
University of Idaho, [email protected]
c
Copyright 2009
The Berkeley Electronic Press. All rights reserved.
A Note on Second Order Probabilities in the
Traditional Deterrence Game∗
Lisa J. Carlson and Raymond Dacey
Abstract
This note focuses on a methodological issue that arises naturally in applications of the traditional deterrence game played under two-sided incomplete information. The problem has potentially interesting implications for the status of the conclusions we draw from various applications
of the traditional deterrence game.
KEYWORDS: deterrence game, second order probabilities
∗
We wish to thank John Symons (University of Texas at El Paso) and the participants in the 2007
Jan Tinbergen European Peace Science Conference for their helpful comments and suggestions.
Carlson and Dacey: Second Order Probabilities in the Traditional Deterrence Game
The Traditional Deterrence Game
Here we use a basic version of the traditional deterrence game to represent a large
class of decision problems commonly encountered in the analysis of conflict. A
more complicated version of the game that encounters the same methodological
issue is employed in Carlson and Dacey (2009). The traditional deterrence game
is presented in Zagare and Kilgour (1993, 2000) and Morrow (1994). In what
follows, we develop a sequential decision analysis of the two-sided incomplete
information version of the traditional deterrence game.
There are two actors in the traditional deterrence game: Challenger and
Defender. Challenger moves first and can choose to either issue a threat or not
issue a threat. If Challenger does not threaten, then the game terminates in the
status quo (SQ ) . If Challenger threatens, then Defender can either resist the threat
or give in to the threat. If Defender gives in, then the game ends in Defender’s
acquiescence ( ACQ ) . If Defender resists, then Challenger can either escalate or
Challenger can back down. If Challenger escalates, then the game ends in war
(WAR ) ; if Challenger backs down, then the game ends in Challenger’s
capitulation (CAP ) . The traditional game under complete information is shown in
Figure 1.
Figure 1 – The Traditional Deterrence Game Under Complete Information
Challenger escalates
WAR
Defender resists
Challenger threatens
CAP
Challenger backs down
ACQ
Defender gives in
SQ
Challenger does not threaten
Published by The Berkeley Electronic Press, 2009
1
Peace Economics, Peace Science and Public Policy, Vol. 15 [2009], Iss. 1, Art. 8
Challenger and Defender can each be one of two types: hard or soft. The
preference orderings for the two types of Challengers and Defenders are as
follows:
Hard Challenger:
Soft Challenger:
Hard Defender:
Soft Defender:
ACQ f SQ f WAR f CAP
ACQ f SQ f CAP f WAR
CAP f SQ f WAR f ACQ
CAP f SQ f ACQ f WAR
In a game of two-sided incomplete information, each player is uncertain about the
other player’s type. Specifically, Challenger has probabilities p and 1 − p that
Defender is hard and that Defender is soft, respectively, and Defender has
conditional probabilities q and 1 − q that Challenger is hard and that Challenger is
soft, given that Challenger has threatened, respectively. Furthermore, Challenger
holds conditional probabilities that Defender resists and does not resist given that
Defender is hard, and that Defender resists and does not resist given that Defender
is soft. The tree for the two-sided incomplete information version of the
traditional deterrence game is presented in Figure 2.
The analysis of a two-sided incomplete information game played as a
sequential decision problem requires counterfactual reasoning within the usual
unfolding process. Defender evaluates nodes 1 and 2 of Figure 2 as if Challenger
were hard or soft, respectively. Thus, at the nodes marked 1, Defender expects a
hard Challenger to escalate, whereas at the nodes marked 2, Defender expects a
soft Challenger to back down. Hence, Defender evaluates the nodes marked 1 as
WAR, and the nodes marked 2 as CAP. At nodes 3 and 4, Defender resists if and
only if qV (WAR ) + (1 − q )V (CAP ) > V ( ACQ ) , where V is Defender’s valuation
function.
At node 3, Challenger expects a hard Defender to resist since
qV (WAR ) + (1 − q )V (CAP ) > V ( ACQ ) for all values of q , and, therefore,
Challenger’s conditional probability that Defender resists, given that Defender is
hard, is unity. However, at node 4, Challenger expects a soft Defender to resist if
and only if qV (WAR ) + (1 − q )V (CAP ) > V ( ACQ ) , which holds only for specific
values of q , and, therefore, Challenger’s conditional probability that Defender
resists, given that Defender is soft, is Challenger’s probability that
qV (WAR ) + (1 − q )V (CAP ) > V ( ACQ ) . Let P denote this probability. Then, by the
total probability rule, Challenger’s probability that Defender resists is
p (1) + (1 − p )(P ) = p + (1 − p )P .
http://www.bepress.com/peps/vol15/iss1/8
DOI: 10.2202/1554-8597.1179
2
Carlson and Dacey: Second Order Probabilities in the Traditional Deterrence Game
Figure 2 – The Traditional Deterrence Game Under Two-Sided Incomplete
Information
Challenger escalates
WAR
Challenger is hard
1
q
CAP
Challenger backs down
Defender resists
Challenger escalates
WAR
Challenger is soft
2
Defender is hard
1-q
3
p
CAP
Challenger backs down
ACQ
Defender gives in
Challenger threatens
Challenger escalates
WAR
Challenger is hard
1
q
CAP
Challenger backs down
Defender resists
Challenger escalates
WAR
Challenger is soft
2
Defender is soft
1-q
4
1-p
CAP
Challenger backs down
ACQ
Defender gives in
SQ
Challenger does not threaten
If Defender resists, then a hard Challenger escalates, and the game ends in WAR,
whereas if Defender resists, then a soft Challenger backs down, and the game
ends in CAP . For convenience, we examine capitulation, and therefore we need
consider only a soft Challenger. Thus, Challenger faces the decision problem
presented in Figure 3.
Published by The Berkeley Electronic Press, 2009
3
Peace Economics, Peace Science and Public Policy, Vol. 15 [2009], Iss. 1, Art. 8
Figure 3 – The Decision Problem for a Soft Challenger Under Two-Sided
Incomplete Information
The payoff table for a soft Challenger’s decision problem is:
Threaten Not Threaten
CAP
SQ
Defender Resists
Defender does not Resist
ACQ
SQ
Probability
p + (1 − p )P
(1 − p )(1 − P )
Challenger will threaten if and only if
( p + (1 − p )P )v(CAP ) + (1 − p )(1 − P )v( ACQ ) > v(SQ )
(1)
where v is Challenger’s valuation function. This inequality rearranges to the
following:
p + (1 − p )P v( ACQ ) − v(SQ )
<
(1 − p )(1 − P ) v(SQ ) − v(CAP )
We refer to the ratio
(2)
p + (1 − p )P
as the probability surface and to the ratio
(1 − p )(1 − P )
http://www.bepress.com/peps/vol15/iss1/8
DOI: 10.2202/1554-8597.1179
4
Carlson and Dacey: Second Order Probabilities in the Traditional Deterrence Game
v( ACQ ) − v(SQ )
as the threshold. The former is an everywhere increasing
v(SQ ) − v(CAP )
function of both p and P , and the latter is a positive number. The graph of the
probability surface and the threshold is presented in Figure 4.
Figure 4 – The Probability Surface and the Threshold
p + (1 − p)P
(1 − p)(1 − P )
threshold = 2
P
p
Published by The Berkeley Electronic Press, 2009
5
Peace Economics, Peace Science and Public Policy, Vol. 15 [2009], Iss. 1, Art. 8
Figure 5 – Graph of the Arc in p − P Space for
v( ACQ ) − v(SQ )
=2
v(SQ ) − v(CAP )
The probability surface and the threshold intersect in an arc. Figure 5 presents the
arc when the threshold is equal to 2. Challenger chooses threaten if and only if the
foregoing inequality holds, i.e., if and only if the pair 〈 p, P〉 falls inside the arc
shown in Figure 5.
Second Order Probabilities
The two-sided incomplete information version of the traditional deterrence game
raises the issue of second order probabilities. As noted earlier, Challenger holds
the probability P that a soft Defender’s expected valuation of resisting is greater
than Defender’s valuation of giving in. That is, P is Challenger’s probability that
qV(WAR) + (1-q)V(CAP) > V(ACQ). Thereby, P is a second order probability,
i.e., a probability over a probability, here q . Second order probabilities cause
serious difficulties for the interpretation of probability theory that is relevant to
the analysis of the traditional deterrence game.
There are three (primary) interpretations of probability theory: the
objective interpretation, wherein probability is a frequency (Bernoulli 1713); the
http://www.bepress.com/peps/vol15/iss1/8
DOI: 10.2202/1554-8597.1179
6
Carlson and Dacey: Second Order Probabilities in the Traditional Deterrence Game
logical interpretation, wherein probability is a logical relation, like deducibility,
between propositions (Keynes 1921, Carnap 1950); and the subjective
interpretation, wherein probability is a personal degree of belief and is measured
as a decision maker’s betting odds (Ramsey 1926, de Finetti 1937, Savage 1954).
Only the subjective interpretation of probability is applicable to the decisionmaking settings of the kind we encounter in the foregoing analysis of the
traditional deterrence game.
The standard account of the subjective interpretation of probability is
based upon the Dutch book argument (Ramsey 1926, de Finetti 1937). A Dutch
book is a set of bets that the decision maker accepts but is a sure-thing loss for the
decision maker. The Dutch book argument is straightforward: if a decision
maker’s betting odds do not obey the laws of probability theory, then there exists
a Dutch book against the decision maker. Jeffrey (2004, 5-6) presents an example
of a Dutch book. In order for second order probabilities to satisfy the subjective
interpretation of probability theory, we must be able to establish that there are no
Dutch books against the decision maker. However, second order probabilities are
such that no events exist against which the bets can be settled. Thus, second order
probabilities encounter a difficulty – the events needed to support the subjective
interpretation of probability theory are not available. Smets (1997, 243) puts the
point as follows:
Second-order probabilities, i.e. probabilities over probabilities, do
not enjoy the same support as subjective probabilities. Indeed,
there seems to be no compelling reason to conceive a second-order
probability in terms of betting and avoiding Dutch books. So the
major justification for the subjective probability modeling is lost.
Further, introducing second-order probabilities directly leads to a
proposal for third-order probabilities that quantifies our
uncertainty about the value of the second-order probabilities. Such
iteration leads to an infinite regress of meta-probabilities that
cannot be easily avoided.
The question then becomes: What interpretation of probability theory is satisfied
by the second order probabilities encountered in the analysis of the traditional
deterrence game? Marshak (1975,1 121) frames the question in terms of the
meaning of second order probabilities, as follows:
1
The paper referred to as Marschak, et al. (1975) is composed of an introductory note by Jacob
Marschak, the outline of a presentation by Morris H. DeGroot, a brief paper by Jacob Marschak
(entitled “Do Personal Probabilities of Probabilities Have an Operational Meaning?”) and
comments by Karl Borch, Herman Chernoff, Morris H. DeGroot, Robert Dorfman, Ward
Edwards, T. S. Ferguson, Koichi Miyasawa, Paul Randolph, L. J. Savage, Robert Schlaifer, and
Published by The Berkeley Electronic Press, 2009
7
Peace Economics, Peace Science and Public Policy, Vol. 15 [2009], Iss. 1, Art. 8
Suppose all probabilities are defined as ‘subjective’, ‘personal’ –
i.e., as being revealed by a ‘rational’, ‘consistent’ decisionmaker’s choices under uncertainty: if he prefers to bet on one
rather than on another event, the former is said to be the
subjectively more probable one. Moreover, a few rather plausible
quasi-logical postulates of ‘rationality’ of choices imply that such
‘subjective probabilities’ have indeed the properties of
‘mathematical measure’. Our question is: what meaning, if any,
can be assigned to the probability of a hypothesis, law, theory that
is itself a probability distribution so that its falsity or truth is not,
in general, an observable event on which bets can be made and
paid-off.
Here, the “hypothesis … that is itself a probability distribution” is the proposition
qV (WAR ) + (1 − q )V (CAP ) > V ( ACQ ) . Challenger does not know the truth-status
of this proposition and assigns to it the probability P. More importantly, and to
Marschak’s point, the proposition is such “that its falsity or truth is not, in general,
an observable event on which bets can be made and paid-off”. Thus, the foregoing
inequality is the kind of proposition Marschak finds troublesome.
Marschak addresses the issues at hand via the question, “Do probabilities
of probabilities have an operational meaning?” (Marschak 1975, 127-133). The
term ‘operational’ has (at least) two uses. As used by Marschak, ‘operational’
refers to the existence of a formal system whereby second order probabilities have
a coherent interpretation as subjective probabilities. The foregoing discussion
shows that second order probabilities cannot be interpreted as subjective
probabilities. As used in the empirical testing of theories and models,
‘operational’ refers to the existence of a practical system whereby values can be
assigned to second order probabilities on the basis of available data. Various
analyses show that practical systems do exist whereby second order probabilities
can thus be interpreted as probabilities. For example, Carlson (1995, 525)
characterizes a second order probability of escalation in terms of military
capabilities in order to test hypotheses drawn from an analysis of escalation based
on the traditional deterrence game.
We are left with an interesting two-part conclusion. Given that the
probabilities we employ in the traditional deterrence game make sense only as
subjective probabilities, the second order probabilities we encounter in the
Robert L. Winkler. The collection is followed by Borch (1975). de Finetti did not participate in
Marschak (1975), but he later responded in de Finetti (1977). de Finetti also addressed the issue of
second order probabilities in de Finetti (1972, 189).
http://www.bepress.com/peps/vol15/iss1/8
DOI: 10.2202/1554-8597.1179
8
Carlson and Dacey: Second Order Probabilities in the Traditional Deterrence Game
foregoing analysis do not have an operational meaning. That is, we do not have a
foundation for interpreting second order probabilities, as encountered in the
traditional deterrence game, as betting odds and thereby as subjective
probabilities. Thus, under Marschak’s use of ‘operational’, second order
probabilities do not have an operational meaning. Contrariwise, under the second
use of ‘operational’, when assessed in terms of measurable factors such as military
capabilities, second order probabilities do have an operational meaning.
It is interesting to note that the purely formal (i.e., axiomatic) aspects of
second-order probabilities have been addressed. Gaifman (1988) provides a
formal structure for higher order probabilities, and Nau (2006) applies secondorder probabilities in an analysis similar to the analysis of the traditional
deterrence game employed here. Nonetheless, second-order probabilities, as
subjective probabilities, remain problematic for the reasons discussed in Marschak
(1975) and highlighted in Smets (1997).
References
BERNOULLI, J., (1713), Ars Conjectandi, Basel.
BORCH, K. (1975), Probabilities of Probabilities, Theory and Decision, vol. 6, n.
2, pp. 155-159.
CARLSON, L. J. (1995), A Theory of Escalation and International Conflict, Journal
of Conflict Resolution, vol. 39, n. 3, pp. 511-534.
CARLSON, L. J., DACEY, R. (2009), The Assassin and the Donor as Third Players
in the Traditional Deterrence Game, The Economics of Peace and Security
Journal, vol. 4, n. 2, pp. 14-22.
CARNAP, R. (1950), The Logical Foundations of Probability, Chicago: University
of Chicago Press.
DE FINETTI, B. (1977), Probabilities of Probabilities: A Real Problem or a
Misunderstanding, pp. 1-10 in Ahmet Aykaç and Carlo Brumat , editors, New
Developments in the Applications of Bayesian Methods, Amsterdam: NorthHolland Publishing Company.
DE FINETTI, B. (1972), Probability, Induction and Statistics, New York: Wiley &
Sons, Inc.
DE FINETTI, B. (1937), La Prevision: Ses Lois Logiques Ses Sources Subjectives,”
Annals de l’ Institut Henri Poincaré, vol. 7, pp. 1-68. (Available in English
translation as “Foresight: Its Logical Laws, Its Subjective Sources,” pp. 99158 in Henry E. Kyburg, Jr. and Howard E. Smokler, editors, Studies in
Subjective Probability, New York: J. Wiley & Sons, Inc., 1964.)
GAIFMAN, H. (1988), A Theory of Higher Order Probabilities, pp. 191-219 in
Skyrms, B., Harper, W. L., editors, Causation, Chance, and Credence, Vol. 1,
Dordrecht, Holland: Kluwer Academic Publishers.
Published by The Berkeley Electronic Press, 2009
9
Peace Economics, Peace Science and Public Policy, Vol. 15 [2009], Iss. 1, Art. 8
JEFFREY, R. C. (2004), Subjective Probability, Cambridge: Cambridge University
Press.
KEYNES, J. M. (1921), A Treatise on Probability, London: Macmillan & Co.
MARSCHAK, J., ET AL. (1975), Personal Probabilities of Probabilities, Theory and
Decision, vol. 6, n. 2, pp. 121-153.
MORROW, J. D. (1994), Game Theory for Political Scientists, Princeton: Princeton
University Press.
NAU, R. F. (2006), Uncertainty Aversion with Second-Order Probabilities,
Management Science, vol. 52, n. 1, pp.136-145.
RAMSEY, F. P. (1926), Truth and Probability, pp. 63-92 in Henry E. Kyburg, Jr.
and Howard E. Smokler, editors, Studies in Subjective Probability, New
York: J. Wiley & Sons, Inc., 1964, and pp. 156-198 in R. B. Braithwaite,
editor, The Foundations of Mathematics and Other Logical Essays, London:
Routledge and Kegan Paul, 1931.
SAVAGE, L. J. (1954), The Foundations of Statistics, New York: Wiley & Sons,
Inc.
SMETS, P. (1997), Imperfect Information: Imprecision and Uncertainty, pages
225-254 in Amihai Motro and Philippe Smets, editors, Uncertainty
Management in Information Systems, Dordrecht, Holland: Kluwer Academic
Publishers.
ZAGARE, F. C., KILGOUR, D. M., (1993), Asymmetric Deterrence, International
Studies Quarterly, vol. 37, pp. 1-27.
ZAGARE, F. C., KILGOUR, D M. (2000), Perfect Deterrence, Cambridge:
Cambridge University Press.
http://www.bepress.com/peps/vol15/iss1/8
DOI: 10.2202/1554-8597.1179
10