Bayesianism Without Priors,
Acts Without Consequences
Robert Nau
Fuqua School of Business
Duke University
ISIPTA 2005
1
Possible sources of imprecision in
subjective (epistemic) probabilities
1. Preferences are incomplete or only partially revealed, so
beliefs are represented by convex sets of probabilities.
2. Preferences are complete but violate the independence
axiom, so beliefs are represented by non-additive or
second-order or otherwise generalized probabilities.
3. Preferences are complete and satisfy independence, but
beliefs are entangled with state-dependent utilities, so
they are represented by risk neutral probabilities
(marginal betting rates for money).
2
Does source #3 matter? Yes!
• Inseparability of probabilities and utilities is potentially
very problematic for game theory (common knowledge
of utilities? common prior probabilities??)
• Also problematic for economic models which try to
separate the effects of “information” and “tastes”
• Also problematic for characterizations of aversion to
risk and/or uncertainty that refer to “true” expected
values or “constant” acts as benchmarks
• And it’s also problematic for the measurement of
previsions!
3
How are previsions measured?
• De Finetti’s method: the lower prevision of x is the
quantity of money you are willing to pay to receive $x.
– But what if utility for money is nonlinear??
• Walley/De Cooman’s method: the lower prevision of x is
the quantity of utility you are willing pay to receive x
utiles.
– But how are “utiles” measured, and what if utility is
state-dependent?
The “utile” problem cannot be finessed away, because...
4
How are utilities measured?
• The authority for a subjective scale of utility comes from
Savage (1954) and Anscombe-Aumann (1963).
• Their models assume the existence of a set of primitive
consequences (“states of the person”).
• Acts are arbitrary, often-counterfactual, mappings of
states to consequences (or objective lotteries thereon).
• The utility of act x is defined as the number  such that x
is indifferent to a lottery that yields the best and worst
consequences with objective probabilities  and 1.
• This definition depends on the conventional assumption
that consequences have state-independent utility, but...
5
State-independence of utility is
inherently untestable
• It is unreasonable to expect that, in all situations, it will be
possible to define “consequences” or “prizes” so that their
utilities will be a priori state-independent—particularly in
situations that people care about.
• The usual axioms of state-independence (Savage’s P3 or
Anscombe-Aumann’s monotonicity) anyway do not rule
out the possibility of state-dependent utility scale factors.
• Aumann’s 1971 letter to Savage: what if the event in
question is the survival of your beloved spouse or the
debunking of the theory on which your life’s work is
6
based? (or the performance of your retirement portfolio...)
This paper:
• Axiomatizes a simple additive representation of preferences
without explicit consequences
• The only primitives are discrete “moves” of two players—
the DM and nature—and monetary gambles
• Implicitly every combination of a move of the DM, a move
of nature, and a monetary payoff, is treated as a distinct
consequence
• Doesn’t yield “true” probabilities—but doesn’t need to!
• Provides a sufficient foundation for Bayesian methods of
decision analysis and game theory based on no-arbitrage
and risk neutral probabilities, which are marginal previsions
(local betting rates measured in terms of money, not utiles)
7
Modeling framework
• Let A = {a1, …, aI} denote a finite set of strategies
(alternative courses of action)
• Let S = {s1, …, sJ} denote a finite set of states. Non-empty
subsets of S are events.
• Let w = {w1, …, wJ}  W  J denote an observable
allocation of money over states (i.e., a side-gamble)
• An act is a strategy-allocation pair (a, w)
• An outcome is a triple (a, s, w)
This is essentially the Arrow-Debreu state-preference
framework augmented by a finite set of strategies
8
Preference axioms
Axiom 1: (Ordering)  is a weak order, and for every
strategy the restricted preference relation over
allocations is a continuous weak order.
Axiom 2: (Strict monotonicity) w*j > wj 
(a, w*j wj)  (a, w) for all a, j, w, w*j.
(More money preferred to less, and no null states)
9
Axiom 3: (Strategic generalized triple cancellation)
For every pair of strategies a and b, allocations w, x, y, z,
state j, and constants wj*, xj*:
If (a, w)  (b, x) and (a, w*j wj)  (b, x*j x-j) and
(a, wj yj)  (b, xj,zj) then (a, w*j yj)  (b, x*j zj).
In words: if changing wj to wj* is no worse than changing xj
to xj* when the background is a comparison of (a, w) vs.
(b, x) , then changing wj to wj* cannot be strictly worse
than changing xj to xj* when different allocations y and z
are received in the other states.
Axiom 4: (Overlap among strategies) There exist
allocations {w(a), aA } in the interior of W such that
(a, w(a)) ~ (b, w(b)) for all strategies a and b.
10
Main result: general additive model
Theorem 1: Axioms 1-4 hold iff  is represented by a
cardinal utility function U having the strategy-dependent
additive-across-states form:
U(a, w) =

sS
vas ( ws ) ,
in which {vas} are strictly increasing, continuous
evaluation functions for money that are unique up to joint
transformations of the form vas + s+as where >0 and,
for every a,   as  0 .
sS
Note: preferences among strategies without any sidegambles are represented by {s vas(0)}.
11
Special cases of the general additive model
1.
SEU with quasi-linear utility
vas(ws) = ps(uas+ws)
2.
SEU with state-dependent linear utility
vas(ws) = ps(uas+sws)
3.
SEU with state-independent utility and prior stakes
vas(ws) = psu(ws+Ws)
4.
General state-dependent SEU
vas(ws) = psuas(ws)
12
Which SEU preference parameters are
theoretically observable ?
• The DM’s “true” subjective probabilities for all events are
observable only in special case 1 (quasi-linear utility).
• True probabilities are distorted by state-dependence and/or
effects of prior stakes in cases 2 and 3.
• True probabilities are irrelevant in special case 4
(hopelessly entangled with utilities).
• But… so what ! True probabilities are not needed for
analysis of decisions (or games).
13
What’s really observable via finite
measurements?
• The decision maker’s preference order has an additive
representation in terms of evaluation functions {vas},
which are infinite-dimensional vectors.
• What sort of finite-dimensional measurements of
preferences can be performed to enable decision analysis
& game-theoretic analysis?
• How can we tell if the decision maker is acting
“rationally” based (only) on this information?
14
Analysis of a finite decision problem
• Henceforth, assume that only a finite number M of concrete
acts are available, accompanied by “small” side-gambles.
• W.l.o.g. let each concrete act be considered as a unique
strategy with zero allocation, e.g., “strategy m” means act
(am, 0) for m = 1, …, M
• An outcome of the decision problem is a pair (m, s).
• Let vms(w) henceforth denote the evaluation function for
money in outcome (m, s).
• How can we predict the decision maker’s choice of m via
credible elicitation of information about her “beliefs” and
“values” encoded in {vms(w)} ?
15
De Finetti’s excellent idea
• Let beliefs (subjective probabilities) be revealed through
offers to accept small gambles.
• Let rationality be defined as the avoidance of arbitrage
(“no Dutch books”).
• Under-appreciated virtue: this approach not only defines
beliefs and rationality, it renders them common knowledge
in a practical sense.
• It can also be extended to the revelation of values (albeit
with imperfect separation of beliefs/values).
• It also provides a natural bridge to game theory and asset
16
pricing theory
Measurable parameters of belief:
risk neutral probabilities
• The decision maker’s risk neutral (betting) probabilities
given strategy m are the normalized derivatives of the
evaluation functions {vms}, evaluated at w=0:
 ms 
 (0)
vms


v
(0)
ms
k 1
n
• Note that they are, in general, act-dependent
• Conditional on the “event” that strategy m is chosen, for
whatever reason, z is a utility non-decreasing “belief
gamble” iff 0  z · m  E [z]
17
Common knowledge of beliefs (CKB)
• Definition: vectors {zk} are acceptable gambles if the DM
is willing to let an opponent choose small non-negative
multipliers {k} and receive a total payoff of k kzk from
the opponent
• Axiom CKB: for each m, the vectors z = ±(1s ms) are
acceptable given strategy m, where 1s is the indicator for
state s and ms is the risk neutral probability of state s given
given strategy m.
• I.e., the DM is willing to buy/sell Arrow-Debreu securities
on state s at price ms in the event she chooses strategy m.
18
In what sense do risk neutral
probabilities {m } represent “beliefs”?
• They coincide with hypothetical “true” probabilities (only)
in the special case of quasi-linear utility
• In all other cases, they are strictly more useful (e.g., they
suffice to determine marginal prices and risk premia)
• They are updated via Bayes’ rule on receipt of new
experimental information, exactly like “true” probabilities
• Individuals ought to eventually agree on risk neutral
probabilities, not “true” probabilities (given the
opportunity to gamble with each other)
19
Measurable parameters of value:
“value gambles”
• In the event strategy m yields greater utility than
strategy n, the gamble mn defined by
vms (0)  vns (0)
 mns 
 (0)
vms
whose state-by-state increments of cardinal utility are
proportional to the state-by-state differences in value
between m and n, evaluated from the perspective of
the local marginal values for money that apply under
the chosen strategy m, is utility non-decreasing. 20
In what sense do {mn} represent “values”
• They coincide exactly with vectors of utility differences
in the special case of quasi-linear utility
• They are independent of beliefs (subjective probabilities)
in the case of state-dependent SEU
• They are independent of information in the case of
scientific experiments
• Axiom CKV: for every strategy m and alternative
strategy n, the vector mn is an acceptable gamble.
21
Decision analysis in terms of
acceptable gambles
• By construction:
m mn  U(am, 0)  U(an, 0).
• Hence the direction of preference between any two
concrete acts can be determined from observations of
acceptable gambles under CKB and CKV.
• This suffices to determine the optimal act, even though
“true” probabilities and utilities have not been observed
• But there is also another way to determine the “rational”
outcome of the decision problem...
22
Common knowledge of rationality
Definition: There is ex post arbitrage in outcome (m, s)
if there exists a non-negative linear combination of
acceptable gambles whose total payoff to the decision
maker[s] is non-positive in all outcomes and strictly
negative in state (m, s)
Axiom CKR: There is no ex post arbitrage.
• In other words, if there is ex post arbitrage in outcome
(m, s), then it should not happen.
23
Solution of the decision problem “by
arbitrage”
• Fundamental theorem of decision analysis [one DM vs.
nature]: Given axioms 1-4, CKB, CKV, and CKR, the
outcome will be one in which the DM chooses a utilitymaximizing strategy and nature chooses a state that has
positive risk neutral probability given that strategy
• Fundamental theorem of games [2 or more DM’s vs.
each other]: Given axioms 1-4, CKV, and CKR, the
outcome will be one that has positive probability in a
subjective correlated equilibrium based on common prior
risk neutral probabilities
24
Onward to Bayesian updating...
• The paper goes on to assume that the state space S can be
partitioned as S = H  I where I is a set of “informational”
events that are expected to be resolved before an act is
chosen and H is a set of “hypotheses” (consequential events)
expected to be resolved only after an act has been chosen.
• Conditional preferences are defined in terms of contingent
strategies that are pegged (only) to events in I.
• The DM is assumed to have no intrinsic interest in I-events,
given the hypotheses, implying an observable likelihood.
• Optimal posterior decisions are made by updating risk
neutral probabilities according to Bayes’ rule, then applying
them to the evaluation of the original value gambles. 25