Charges and Bets: A General Characterisation ofCommon Priorsjoint with Ziv Hellman

Miklós Pintér

Corvinus Center for Operations Research

Mathematical Game Theory Seminar 2020

The finite case (Samet, 1998)

Definition: A type space for a set of agents I = {1, . . . , n} is atuple (Ω, (Πi , ti )i∈I ), where

I Ω is a finite set of states,

I Πi is a partition of Ω, and

I ti is a function ti : Ω→ ∆(Ω), which associates with eachstate ω the type of i at ω, i.e., a point in ∆Ω, the simplex inRΩ, which we consider as the set of probability distributionsover Ω. The type function ti satisfies the following twoconditions:

I for each ω ∈ Ω, ti (ω)(Πi (ω)) = 1, where Πi (ω) is the elementof the partition Πi which contains ω,

I ti is constant over each element of Πi .

The finite case (Samet, 1998) I

Example: Consider the following type space (Ω, (Πi , ti )i∈I ), where

I the state space is Ω = {ω1, ω2, ω3, ω4},I the players set is I = {1, 2},I the two partitions are Π1 = {{ω1, ω2}, {ω3, ω4}} and

Π2 = {{ω1, ω3}, {ω2, ω4}},I the two type functions t1 : Ω→ ∆(Ω) are such that

t1(ω1) = t1(ω2) =

(1

2,

1

2, 0, 0

), (1)

and

t1(ω3) = t1(ω4) =

(0, 0,

1

2,

1

2

). (2)

Moreover, let t2 : Ω→ ∆(Ω) be such that

The finite case (Samet, 1998) II

t2(ω1) = t2(ω3) =

(1

2, 0,

1

2, 0

), (3)

and

t2(ω2) = t2(ω4) = (0, 1, 0, 0) . (4)

The finite case (Samet, 1998)

Definition: Take a type space (Ω, (Πi , ti )i∈I ). Then the probabilitydistribution Pi ∈ ∆(Ω) is a prior for player i , if for each πi ∈ Πisuch that Pi (πi ) > 0 it holds that ti (ω)(·) = Pi (· | πi ), ω ∈ πi . LetPi denote the set of player i ’s priors.

A probability distribution P is a common prior, if P ∈ ∩i∈IPi , inwords, a common prior is a prior for every player.

The finite case (Samet, 1998)Symbolically, we can partition the space of type spaces T over Ωepistemically into consistent type spaces C and inconsistent ones I

T

C

I

The finite case (Samet, 1998)

Example: Considering the type space above, it does not attain acommon prior. Suppose for contradiction that P is a commonprior, then by Equation (1)

P({ω1}) = P({ω2})

and by Equation (3)

P({ω1}) = P({ω3})

and by Equation (2)

P({ω3}) = P({ω4})

which contradicts Equation (4) (P({ω2}) 6= P({ω4})).

The finite case (Samet, 1998)

Definition: A collection of functions (fi )i∈I , fi : Ω→ R, i ∈ I , is abet for the type space (Ω, (Πi , ti )i∈I ), if

∑i∈I fi = 0. The bet

(fi )i∈I is an agreeable bet if for each i ∈ I and ω ∈ Ω it holds that∫fi d ti (ω) > 0.

The finite case (Samet, 1998)

We can partition T over Ω behaviourally into type spaces thatadmit agreeable bets B and ones that do not D

T

D

B

The finite case (Samet, 1998)

Example: Considering the type space above, let

f1(ω) =

2 if ω = ω1,−1 if ω = ω2,−3 if ω = ω3,4 if ω = ω4.

Moreover, let f2 = −f1. Then (f1, f2) is an agreeable bet.

The finite case (Samet, 1998)

Theorem: A type space (Ω, (Πi , ti )i∈I ) attains a common prior ifand only if there does not exist an agreeable bet for it.

Let P1,P2 be respectively the (compact and convex) sets of priorsof player 1 and player 2

P1

P2

The finite case (Samet, 1998)The theorem above tell us that we get the same partition of thespace of type spaces in both cases: C = D

T

C = D

I = B

The problem we address

I What can we say in the infinite setting?

I Can we generalize Samet (1998)’s characterization to theinfinite setting?

I If yes, then how?

Motivation

I Aumann (1976)’s No Disagreement, or No Betting Theorem,

I No Betting Theorem (proved independently by severalresearchers in 1990s),

I This has had broad implications in the study of speculativetrade, interactions between Bayesian agents, Bayesianpersuasion, and many other fields,

I CPA (Common Prior Assumption) is central in economictheory, but how to test CPA?

Feinberg (2000)’s counterexample

Anne 1 2313

12

12

23

13

12 . . .

Ben 1212

12

12

12

12

12

12

12 . . .

In this type space:

I There is no common prior,

I There is no agreeable bet either.

Hopes (positive results)

I By Feinberg (2000) and Heifetz (2006) ifI The state space is compact,I The type functions are continuous,I The beliefs are (compact) regular probability measures,I The bets are continuous functions,

then the epistemic and behavioral equivalence is restored: Atype space attains a common prior if and only if there doesnot exist an agreeable bet for it.

I For countable type spaces, Lehrer and Samet (2014) present athree-levelled epistemic classification – weakly beliefconsistent, belief consistent, strongly belief consistent –equivalent to three behavioural properties.

What to generalize?

Questions to address:

I What should be the state space? Options: a topologicalspace, a measurable space, etc.,

I What should be the beliefs? Options: probability measures,regular probability measures, probability charges, etc.,

I What should be the type functions? Options: continuousfunctions, measurable functions, some integrable functions,etc.,

I What should be the priors? Options: ’prior-first’ or’posterior-first’,

I What should be the bets? Options: bounded continuousfunctions, bounded measurable functions, some integrablefunctions, etc.

Brief Review of Finite Additivity Concepts

I If A is a field on Ω then (Ω,A) is called a chargeable space,I A set function µ : A → R on a chargeable space is a

probability charge if µ is non-negative, additive, andµ(Ω) = 1,

I If A is a σ-field then (Ω,A) is a measurable space,I If probability charge µ on a measurable space is σ-additive it

is a probability measure,

I Probability measures are special cases of probability charges.

Integral if the integrator is a charge

Take a charge space (X ,A, µ), and let S(X ,A) denote the class ofstep functions over (X ,A). For a bounded function f : X → Rdefine the lower integral of f by

I∗(f ) = sup

{∫s dµ : s ∈ S(X ,A) and s ≤ f

},

and the upper integral by

I ∗(f ) = inf

{∫s dµ : s ∈ S(X ,A) and s ≥ f

}.

A function f is µ-integrable, if I∗(f ) = I∗(f ).

A function f is A-integrable, if it is integrable by any probabilitycharge over (X ,A).The dual of the A-integrable functions is ba(X ,A), and the set ofpba(X ,A) is weak* compact.

Our model

I We suppose throughout the existence of a a chargeable space(Ω,M).

I The elements of the field M, the general epistemic field, serveas the events of interest in our model.

I The model also includes a set of players N (which is notnecessarily assumed to be a finite set).

I Each player i ∈ N is associated with a field Mi ⊂M (i ’sprivate epistemic field), which together with Ω forms achargeable space (Ω,Mi ).

Our model

Definition: A type function is a mapping ti : Ω×M→ [0, 1]satisfying

1. ti (ω, ·) is a probability charge on M for all ω ∈ Ω,2. ti (·,E ) is Mi -integrable for each event E in the field M.

Definition: A tuple ((Ω,M), {(Ω,Mi )}i∈N , {ti}i∈N) is called atype space if for each player i ∈ N, player i ’s type function tisatisfies the property that for each E ∈Mi , for each ω ∈ E ,ti (ω,E ) = 1.

Our modelDefinition: Let T = ((Ω,M), {(Ω,Mi )}i∈N , {ti}i∈N) be a typespace, and let i ∈ N be a player. Then a probability chargePi ∈ pba(Ω,M) is a prior of player i (relative to T ) if for eachA ∈M and B ∈Mi

Pi (A ∩ B) =∫B

ti (·,A) dPi .

Πi will be used to denote the set of player i ’s priors (with Tunderstood from context).Definition: A charge P ∈ pba(Ω,M) is a common prior, if

P ∈⋂i∈N

Πi .

Lemma: Let ((Ω,M), {(Ω,Mi )}i∈N , {ti}i∈N) be a type space andlet i ∈ N be a player. Then Πi , the set of player i ’s priors, is theweak* closure of the convex hull of i ’s types, that is,

Πi = conv({ti (ω, ·) : ω ∈ Ω})∗.

‘Thickening’

Imagine ‘ε thickening’ the sets of priors by adding more probabilitycharges that are ‘ε distant’ (by the TV norm) from the boundariesof the sets. Denote the resulting sets Πεi

Π1

Π2

ε

ε

Πε1

Πε2

‘Thickening’

Can ask how much ε > 0 thickening until there is an intersection?This intuitively measures ‘how far’ Π1 and Π2 are from each other

Π1

Π2

ε

ε

Πε1

Πε2

‘Thinning’

For ε < 0, imagine ‘thinning’ sets of priors by removing probabilitycharges that are near the boundary

Π1

Π2

‘Thinning’

Can ask how much thinning until Πε1 and Πε2 are disjoint?

Πε1

Πε2

Π1

Π2

Our model

Definition: For each player i and each ε ∈ [−1, 1] define Πεi =(Πi +OTV (0, ε))

∗ ∩ pba(Ω,M) if ε > 0,Πi if ε = 0,

{µ ∈ pba(Ω,M) : OTV (µ,−ε) ∩ pba(Ω,M)) ⊆ Πi}∗

if ε < 0,

A charge P is an ε-common prior for ε ∈ [−1, 1] if

P ∈ ∩i∈NΠεi .

Our model

Definition: Let T = ((Ω,M), {Ω,Mi}i∈N , (ti )i∈N) be a typespace. A set of M-measurable functions f = {fi1 , . . . , fin}, for afinite index set i1, . . . , in ∈ N, is a bet if

∑nm=1 fim = 0.

A bet is an agreeable bet (relative to T ) if there exists α ∈ R suchthat ∫

fim d tim(ω, ·) ≥ α > 0,

for every state ω ∈ Ω and every player im with m ∈ {1, . . . , n}.

Our model

Definition: Let T = ((Ω,M), {Ω,Mi}i∈N , (ti )i∈N) be a typespace. A set of functions f = {fi1 , . . . , fin} ∈ B(Ω,M), for a finiteindex set i1, . . . , in ∈ N, is a bet if

∑nm=1 fim = 0.

A bet is an ε-agreeable bet (relative to T ) for ε ∈ [−1, 1] if thereexists α ∈ R such that∫

fim d tim(ω, ·) ≥ α > ε‖fim‖sup,

for every state ω ∈ Ω and every player im with m ∈ {1, . . . , n}.

The main result

Theorem: Let T = ((Ω,M), {Ω,Mi )}i∈N , {ti}i∈N) be a typespace and ε ∈ [−1, 1]. Then only one of the following two cases ispossible:

I T admits an ε-common prior.

I There exists an ε-agreeable bet.

The proof (for ε = 0)

Theorem: Let K1, . . . ,Kn be non-empty, compact, convex sets in alocally convex topological vector space X such that 0 /∈ Km,m = 1, . . . , n. Then ∩nm=1 cone(Km) = {0}, wherecone(B) = {αx : α ≥ 0, and x ∈ B}, if and only if there existcontinuous linear functionals f1, . . . , fn over X , and α > 0, suchthat fm(x) ≥ α for each m = 1, . . . , n and for all x ∈ Km, and inaddition

∑fm = 0.

The proof (for ε = 0) I

If: Suppose by contradiction that ∩nm=1 cone(Km) 6= {0} and thatat the same time there exist linear functionals f1, . . . , fn such thatfm(x) ≥ α > 0, for each m = 1 . . . , n and for each x ∈ Km, with∑n

m=1 fm = 0. Since ∩nm=1 cone(Km) 6= {0}, there is an x 6= 0such that x ∈ ∩nm=1 cone(Km); such an x then satisfies theproperty that there exist β1, . . . , βn > 0 such that βmx ∈ Km form = 1, . . . , n, and

∑fm(x) ≥ α

∑nm=1

1βm

. This contradicts∑fm(x) = 0.

Only if: Let K̂ = {x ∈ Kn−11 : x1 = . . . = xn−1, x1 ∈ K1) (so thatK̂ is an n − 1 copy of K1), and letK̃ = cone(K2)× . . .× cone(Kn). It is clear that K̃ is weakly*closed and convex and that K̂ is weakly* compact and convex.Suppose that ∩nm=1 cone(Km) = {0}. Then it follows from thedefinitions that K̃ ∩ K̂ = ∅, which implies that there exists acontinuous linear functional g , a real number β, and ε > 0 suchthat g(x) ≥ β + ε for all x ∈ K̃ and g(x) ≤ β − ε for all x ∈ K̂ .

The proof (for ε = 0) II

Since K̃ contains the origin, it must be the case that β + ε ≤ 0,which implies that β < 0. Moreover, by definition of K̃ ,g = g2 + . . .+ gn, and gm(x) ≥ 0 for each x ∈ Km and eachm = 2, . . . , n.Since Km is closed and 0 /∈ Km, m = 2, . . . , n, there exist αm > 0and δm continuous functionals such that αm ≤ δm(x) ≤ −β2(n−1) ,x ∈ Km, m = 2, . . . , n. Let α = min{α2, . . . , αn} > 0.Then (gm + δm)(x) ≥ α > 0 for each m = 2, . . . , n and for eachx ∈ Km. Furthermore,

∑nm=2(gm + δm)(x) ≤

β2 for all x ∈ K̂ .

Note that K1 and K̂ are isomorphic, hence∑nm=2(gm + δm)(x) ≤

β2 for all x ∈ K1.

Finally, for each m = 2, . . . , n let fm = gm + δm. We now have allthe ingredients for defining a zero-sum agreeable bet: letf1 = −

∑nm=2 fm. Then fm(x) ≥ α > 0 for each m = 1, . . . , n and

each x ∈ Km, and∑

fm = 0.

An example I

Let Ω = {ω1, ω2}, N = {1, 2}, t1 = δ{ω1}, t2 = δ{ω2}, M = P(Ω),and Mi = {∅,Ω} for each player i ∈ N. Then only the pair offunctions

f1(ω) =

{α if x = ω1,−β if x = ω2,

(5)

and f2 = −f1 form an agreeable bet for the type space((Ω,M), {Ω,Mi )}i∈N , {ti}i∈N), for all α, β > 0.Let Ωi = Ω× {i} for each i ∈ R, yielding continuum-many copiesof Ω. Let Ω∗ = ∪i∈IΩi , let M∗ be the field generated by the sets{ω}, for ω ∈ Ω∗ (i.e., the coarsest field containing the singletons ofΩ∗), and let

t∗j (ω, {x}) ={

tj((ω|Ω), {x |Ω}) if ∃i ∈ I such that ω, x ∈ Ωi ,0 otherwise,

j = 1, 2.

An example IIDefine M∗j to be the field generated by Ωi , i ∈ I , j ∈ N = {1, 2}.Then T ∗ = ((Ω∗,M∗), {(Ω∗,M∗j )}j∈N , {t∗j }j∈N) forms a typespace.Suppose by contradiction that there exists an agreeable bet (f1, f2)on T ∗. Then for each i ∈ I , the functions f1 and f2 restricted to Ωimust be defined on Ωi as in Equation (5) (the specific values of αand β might depend on i).Hence there exist ε > 0 such that f −11 ([ε,∞)) and f

−11 ((−∞,−ε))

are both unions of infinitely many singletons, meaning that f1 isnot M∗-integrable, hence (f1, f2) cannot be an agreeable bet.As there is no agreeable bet on T ∗, by our Theorem T ∗ admits acommon prior. Indeed, it is easy to check that the probabilitycharge on M∗ which assigns 0 to each finite set is a common prior.Moreover, for any f ∈ B(Ω∗,M∗), for any finite copy of (Ω,M),for each ω′ ∈ Ω′, n ∈ N∣∣∣∣∫ f dtn(ψ(ω′), ·)− ∫ f ◦ ψ d t ′n(ω′)∣∣∣∣ = 0,

An example III

where Ω′ = ∪i∈JΩi , J ⊆ I is an arbitrary nonempty, finite set,M′ =M|Ω′ and t ′n = tn ◦ ψ, i ∈ N, ψ : Ω′ → Ω is such thatψ(ω) = ω, ω ∈ Ω′.In words, T ∗ has many finite, even ‘perfect’, approximations.Despite this, none of these finite approximations admits a commonprior.

Thank you for the attention!

References I

Aumann RJ (1976) Agreeing to Disagree. The Annals of Statistics4(6):1236–1239

Feinberg Y (2000) Characterizing Common Priors in the Form ofPosteriors. Journal of Economic Theory 91:127–179

Heifetz A (2006) The positive foundation of the common priorassumption. Games and Economic Behavior 56:105–120

Lehrer E, Samet D (2014) Belief consistency and tradeconsistency. Games and Economic Behavior 83:165–177

Samet D (1998) Common Priors and Separation of Convex Sets.Games and Economic Behavior 24:172–174

Why Bets Must Be Bounded Away From Zero I

Our main theorem characterises the existence of common priorsusing agreeable bets which are not only give rise to positiveexpectations at each state but have expectations bounded awayfrom zero. We show here by an example why this presumption isnecessary.There are two players, Anne and Ben. The state space andpartition is the basic partition space, that is, Ω = {1, 2, . . .},Anne’s knowledge partition, ΠA, is given by

{{1}, {2, 3}, {4, 5}, {6, 7}, . . .}

and Ben’s knowledge partition, ΠB , is given by

{{1, 2}, {3, 4}, {5, 6}, . . .}.

The epistemic events: M is the field generated by the singletonsets, Mi is the field generated by the partition Πi , i = A,B.Anne’s type function, tA, is given by

Why Bets Must Be Bounded Away From Zero II

tA(n, {n}) =

1 if n = 112 if n is even12 if n is odd, n > 1.

Ben’s type function, tB , is given by

tB(n, {n}) ={

12 if n is odd12 if n is even.

Anne 1 1212

12

12

12

12

12

12

12

12 . . .

Ben 1212

12

12

12

12

12

12

12

12

12 . . .

Figure: The type space of the example in this section.

Why Bets Must Be Bounded Away From Zero III

Notice that the probability charge which assigns zero to each finiteset is a common prior for this type space.Suppose that fA, with fB = −fA, is an agreeable bet, with |fA| abounded and strictly increasing function andfA(n) = (−1)n+1|fA(n)|, n ∈ Ω. For i = A,B, for each n ∈ Ω,∫

fi dti(n, ·) > 0,

but there does not exist α ∈ R such that for each n ∈ Ω∫fi dti(n, ·) ≥ α > 0.

In words, this type space has an agreeable bet in sense of Lehrerand Samet (2014) but it does not have an agreeable bet in senseof our definition.

A lemma I

Lemma: Take the sets P1,P2 ⊆ `1 ⊂ ba(N). Then P1TV

and

P2TV

are strongly separable by a (norm continuous) linearfunctional if and only if P1

∗and P2

∗are strongly separable by a

(weak* continuous) linear functional.

Proof : If: Since PiTV ⊆ P1

∗, i = 1, 2, if P1

∗and P2

∗are strongly

separable by a linear functional, then the very same linear

functional strongly separates P1TV

and P2TV

.

Only if: Suppose that P1TV

and P2TV

are strongly separable by alinear function over `1, and let f be the non-trivial stronglyseparating linear functional, i.e, there exist c ∈ R and ε > 0 suchthat

f (x) ≥ c + ε and c − ε ≥ f (y),

for all x ∈ P1TV

and for all y ∈ P2TV

.

A lemma II

Since f ∈ `∞ it holds that

f (x) ≥ c + ε and c − ε ≥ f (y),

for all x ∈ P1∗

and for all y ∈ P2∗.

The probleWhy is it important?Why is it not trivial?What is the solution?