Game-theoretic Modeling of Players' Ambiguities on ......t n2T n;tn tn;t n (P(R n;tn)) n 14/37. Game-theoretic Modeling of Players’ Ambiguities on External Factors The Distribution-based

Game-theoretic Modeling of Players’ Ambiguities on External Factors

Game-theoretic Modeling of Players’ Ambiguities onExternal Factors

Jian YangDepartment of Management Science and Information Systems

Business School, Rutgers UniversityNewark, NJ 07102

Email: [email protected]

July 2018

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Return-distribution Vector

We consider a game in which players n receive private messages(their types) tn about external factors (states of the world) ω

State space Ω is partitioned into (Ωn,tn)tn∈Tn for each n

When type profile t ≡ (tn)n∈N is eventually revealed, ω will beknown to have come from Ωt ≡

⋂n∈N Ωn,tn

Players adopt potentially random actions based on their types

Knowing his type tn but not others’ types let alone true ω, player nshould anticipate one return distribution say π(ω) per ω ∈ Ωn,tn

He will certainly want to make return-distribution vectorπ ≡ (π(ω)|ω ∈ Ωn,tn) as likeable to himself as possible

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Preference Relation

When there are two players, imagine state space Ω as a rectangle

Each Ω1,t1 can be understood as a row-band and each Ω2,t2 acolumn-band; all Ω1,t1 ’s partition Ω in a row-by-row fashion and allΩ2,t2 ’s do same in a column-by-column fashion

Each Ωt ≡ Ω(t1,t2) ≡ Ω1,t1 ∩ Ω2,t2 will be a sub-rectangle

A return-distribution vector π ≡ (π(ω)|ω ∈ Ω1,t1) faced by(1, t1)-player (player 1 when receiving type message t1) is amapping from row-band Ω1,t1 to distributions of his returns

A natural apparatus to express (n, tn)-player’s taste is a strictpreference relation �n,tn on all return-distribution vectors,indicating strict preferences between pairs of these vectors

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The Traditional Approach

Traditional expected-utility approach uses what we call areal-valued satisfaction function sn,tn on return-distribution vectors:

π �n,tn π′ if and only if sn,tn(π) > sn,tn(π′)

Moreover, with single prior ρn,tn on states and real-valued utilityfunction un,tn defined on return space Rn,tn ,

sn,tn(π) = s0n,tn(π, ρn,tn)

=∫

Ωn,tn{∫Rn,tn

un,tn(r) · [π(ω)](dr)} · ρn,tn(dω)=∫Rn,tn

un,tn(r) · [∫

Ωn,tnπ(ω) · ρn,tn(dω)](dr),

which is both concave and convex (call it linear) in π

Harsanyi (1967-68) took this approach and by specialization tocase where Tn’s are singletons, so did Nash (1950-51)

4 / 37


Critiques of Traditional Approach

Allais (1953) challenged linearity on integrated return distribution∫Ωn,tn

π(ω) · ρn,tn(dω); probably more importantly, Ellsberg (1961)questioned singleton prior set {ρn,tn} for uncertainty over Ωn,tn

Starting from Schmeidler (1989), researchers resorted to tools likecapacities and Choquet integration to help with decision makinginvolving general ambiguity attitudes

For instance, Gilboa and Schmeidler (1989) legitimized

sn,tn(π) = infρ∈Pn,tn

s0n,tn(π, ρ),

a worst-prior (out of Pn,tn) form expressing ambiguity aversion

Failure to account for players’ diverse ambiguity attitudes couldlead to weird predictions or dangerous prescriptions

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Two Equilibrium Concepts

We can already define preference game based on �n,tn relations

Depending on how behavioral strategies are enforced, there can betwo equilibrium concepts—action- and distribution-based

An action-based equilibrium assigns weights only to actions thatleave no room for improvement by any other pure action—imagineplayers can exert direct control on pure actions while having tomaintain them at predetermined frequencies

A distribution-based equilibrium leaves no room for improvementby any other distribution of actions—imagine players use randomseed generators and predetermined seed-to-action mappings togenerate actions, with random seeds verifiable post-game

6 / 37


Summary of Results

Action-based equilibria will exist (Ea 6= ∅) under mild conditions,and distribution-based equilibria will exist (Ed 6= ∅) so long asplayers have ambiguity-averse tendencies

Both Ea and Ed are upper hemi-continuous in players’ ambiguityattitudes—as each �n,tn can be viewed as a subset in space ofpairs formed by return-distribution vectors, distance andconvergence can be defined for preferences

When satisfaction functions sn,tn substantiate relations �n,tn , wehave satisfaction game; when every sn,tn takes worst-prior form ofGilboa and Schmeidler (1989), we have alarmists’ game

We call opposite case enterprising game because players betoptimistically on most favorable resolutions of ambiguities

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Further Developments

We will have Ed ⊆ Ea 6= ∅ when players are ambiguity-seeking,such as in enterprising game; we will have Ed = Ea 6= ∅ whenplayers are ambiguity-neutral—this is why traditional game has noneed to distinguish between the two equilibrium notions

In general, pure distribution-based equilibria are pure action-based

ones (1A ∩ Ed ⊆ 1A ∩ Ea); for enterprising game, 1A ∩ Ed and1A ∩ Ea are unified say at 1E

With strategic complementarities, we will have not only 1E 6= ∅ butalso presence of monotone, pure equilibria

Applications include auction where bidders are ambiguous aboutitem’s worth and price competition with ambiguity on demand

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Ambiguity on External Factors

Normal-form games involving players’ ambiguities aboutopponents’ strategies were studied by Dow and Werlang (1994),Klibanoff (1996), Lo (1996), Epstein (1997), Eichberger andKelsey (2000), Marinacci (2000)

Between external factors and opponents, often former are lessunderstood—think of Stag Hunt where there could be millions ofcombinations about conditions of preys and hunting ground

Behavioral strategies of players can be verified through differentmeans, leading to action- vs. distribution-based distinction

With types conveying incomplete information, ambiguity onexternal factors ω implies ambiguity on opponents’ behaviors aswell as their ambiguity attitudes

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Ambiguity in Incomplete-information Games

Epstein and Wang (1996), Ahn (2007), and Di Tillio (2008)justified emergence of largest necessary type spaces when playershave ambiguities on opponents’ ambiguity attitudes

We suppose preferences �n,tn are commonly known—think ofplayers’ personal traits like “reckless when stake is high” (butactual opponent types are unknown in-game)

Kajii and Ui (2005) effectively studied alarmists’ game withsn,tn(π) = infρ∈Pn,tn

∫Rn,tn

un,tn(r) · [∫

Ωn,tnπ(ω) · ρ(dω)](dr)

Azrieli and Teper (2011) considered a special satisfaction gamewith sn,tn(π) = jn,tn [(

∫Rn,tn

un,tn(r) · [π(ω)](dr)|ω ∈ Ωn,tn)]

We contribute on general game framework, equilibrium existence,continuity, and relationships, as well as enterprising game

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Other Relevant Literature

Grant, Meneghel, and Tourky (2016) proposed Savage game inwhich players possess preferences over strategy profiles made up ofall players’ deterministic action plans

Empowering players with abilities to directly rank strategy profilesmight have overstated actual players’ sophisticated-ness, andshifted too much burden from game’s analysis to its setup

Our focus on returns rather than strategy profiles is likely morerealistic and parsimonious

Riedel and Sass (2014) allowed players to adjust ambiguities aboutrandom devices used in action generations

Ambiguity in mechanism design—Bose and Renou (2014); that inauction—Lo (1998) and Bose, Ozdenoren, and Pape (2006)

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Game Primitives

Players come from finite set N , with private type tn of each playern ∈ N coming from finite space Tn

An (n, tn)-player can choose action an,tn from space An,tn

States of the world come from Ω, with (Ωn,tn)tn∈Tn forming apartition of it for every n ∈ N—after type profile t ≡ (tn)n∈N isrevealed, players will know action profile a to have come fromAt ≡

∏n∈N An,tn and external factor ω from Ωt ≡

⋂tn∈Tn Ωn,tn

Function rn,t ≡ rn,tn,t−n from At × Ωt to space Rn,tn describeshow action and state profiles translate into (n, tn)-player’s return

During game, (n, tn)-player knows only that ω ∈ Ωn,tn

When players mete out behavioral strategies, current one will facechoices on return-distribution vectors π ≡ (π(ω)|ω ∈ Ωn,tn)

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The Action-based Perspective

A player may take a frequentist’s approach to compliance withproclaimed strategy—choice is on pure action an,tn in each play

Under own action an,tn , opponent strategy profileδ−n ≡ (δm,tm)m 6=n,tm∈Tm and type profile t−n, and stateω ∈ Ωtn,t−n , the (n, tn)-player will expect return distribution

πan,tn,t−n(an,tn , δ−n,t−n , ω) =

∏m6=n

δm,tm

·(rn,tn,t−n(an,tn , ·, ω))−1,meaning that, for any measurable subset R′ of Rn,tn ,

[πan,tn,t−n(an,tn , δ−n,t−n , ω)](R′) = (

∏m6=n δm,tm)

({a−n,t−n ∈ A−n,t−n |rn,tn,t−n(an,tn , a−n,t−n , ω) ∈ R′})

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From Action to Vector

Had player n known his opponents’ type profile t−n ∈ T−n, hewould have anticipated “|Ωtn,t−n |-dimensional” vector

πan,tn,t−n(an,tn , δ−n,t−n) ≡(πan,tn,t−n(an,tn , δ−n,t−n , ω)|ω ∈ Ωtn,t−n

)However, since player is unaware of opponents’ actual type profile,he should contemplate on “|Ωn,tn |-dimensional” vector that ispatched up from shorter vectors:

(πan,tn,t−n(an,tn , δ−n,t−n))t−n∈T−n= ((πan,tn,t−n(an,tn , δ−n,t−n , ω)|ω ∈ Ωtn,t−n))t−n∈T−n= (πan,tn,t−n(an,tn , δ−n,t−n , ω)|ω ∈ Ωn,tn)

Resulting return-distribution vector πan,tn(an,tn , δ−n) is a member

of (P(Rn,tn))⋃

t−n∈T−nΩtn,t−n ≡ (P(Rn,tn))Ωn,tn

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The Distribution-based Perspective

It could happen that every player is given a random seed generatorwhose output is private knowledge in-game but public knowledgepost-game, and the player has to act according to an agreed-uponmapping from the random output to his action

Under own strategy δn,tn , opponent strategy profile δ−n and typeprofile t−n, and state ω ∈ Ωtn,t−n , the (n, tn)-player will expect

πdn,tn,t−n(δn,tn , δ−n,t−n , ω) =

δn,tn × ∏m 6=n

δm,tm

·(rn,tn,t−n(·, ·, ω))−1,or in a sense

∫An,tn

πan,tn,t−n(a, δ−n,t−n , ω) · δn,tn(da)

Let πdn,tn(δn,tn , δ−n) stand for distribution-based return-distribution

vector ((πdn,tn,t−n(δn,tn , δ−n,t−n , ω)|ω ∈ Ωtn,t−n))t−n∈T−n15 / 37


Preference on Return-distribution Vectors

Suppose each (n, tn)-player merely possesses a preference relation�n,tn among return-distribution vectors in (P(Rn,tn))Ωn,tn , so that

(I) π 6�n,tn π for any π (irreflexivity);(II) π �n,tn π′′ whenever π �n,tn π′ and π′ �n,tn π′′ (transitivity)

For an (n, tn)-player, let state space Ωn,tn = {hot day, cold day}and return space Rn,tn = {ice cream, beef stew}

A preference relation �n,tn may imply that “ice cream when it ishot and beef stew when it is cold” is more preferable than “eitherwith 50% chance on either type of day”, which is more preferablethan “beef stew when it is hot and ice cream when it is cold”

We can already define preference game based on the �n,tn relations

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Preference Game—Action-based Perspective

Let Âan,tn(δ−n) be set of actions that maximize πan,tn(·, δ−n), i.e.,{

a ∈ An,tn |πan,tn(a′, δ−n) 6�n,tn πan,tn(a, δ−n) ∀a

′ ∈ An,tn}

Let best-response correspondence B̂an,tn be such that, for any δ−n,

B̂an,tn(δ−n) ={δn,tn ∈ P(An,tn)|δn,tn(Âan,tn(δ−n)) = 1

}Define B̂a from

∏n∈N

∏tn∈Tn P(An,tn) to itself, so that

δ′ ∈ B̂a(δ) if and only if δ′n,tn ∈ B̂an,tn(δ−n) ∀n ∈ N, tn ∈ Tn

A behavioral-strategy profile δ ≡ (δn,tn)n∈N,tn∈Tn will be anaction-based equilibrium, i.e., a member of Ea, if δ ∈ B̂a(δ)

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Preference Game—Distribution-based Perspective

Let B̂dn,tn(δ−n) be set of distributions that maximize πdn,tn(·, δ−n):{

d ∈ P(An,tn)|πdn,tn(δ′, δ−n) 6�n,tn πdn,tn(d, δ−n) ∀δ

′ ∈ P(An,tn)}

Define B̂d from∏n∈N

∏tn∈Tn P(An,tn) to itself, so that

δ′ ∈ B̂d(δ) if and only if δ′n,tn ∈ B̂dn,tn(δ−n) ∀n ∈ N, tn ∈ Tn

A behavioral-strategy profile δ ≡ (δn,tn)n∈N,tn∈Tn will be adistribution-based equilibrium, i.e., a member of Ed, if δ ∈ B̂d(δ)

It will be interesting to know when Ea and Ed are nonempty,whether they change continuously with underlying game, and what

their relations (Ed ⊆ Ea or the other way around, etc.) look like

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Continuity and Compactness Assumptions

Assume that all action spaces An,tn , state space Ω, and all returnspaces Rn,tn are compact

Assume that sets Ωt ≡⋂n∈N Ωn,tn are closed hence compact

This guarantees “separability” in sense of

dΩ(Ωt,Ωt′) > 0 ∀t 6= t′

Assume that return functions rn,tn,t−n(·, ·, ·) are continuous

Assume that each �n,tn is continuous, so that π �n,tn π′ isextend-able to small neighborhoods around π and π′

An immediate consequence is existence of maximal element π inany compact subset Π′ of return-distribution vectors, so thatπ′ 6�n,tn π for any π′ ∈ Π′

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Existence of Equilibria

Then, πan,tn maps continuously from own actions and opponents’strategy profiles to continuous return-distribution vectors

This makes correspondence Âan,tn both nonempty and closed,

further leading to nonemptiness and closedness of B̂an,tn

Using Fan-Glicksberg fixed point theorem, we can show thataction-based equilibria exist for preference game: Ea 6= ∅

Similarly, πdn,tn maps continuously to continuous return-distribution

vectors, so that correspondence B̂dn,tn is nonempty and closed

When �n,tn is convex so that π′ 6�n,tn π0 and π′ 6�n,tn π1 wouldensure π′ 6�n,tn (1− α) · π0 + α · π1 for α ∈ [0, 1],distribution-based equilibria will exist: Ed 6= ∅

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Space of Players’ Ambiguity Attitudes

Preferences �n,tn can be identified with nonempty closed sets

ϕn,tn = {(π, π′) ∈ Πn,tn ×Πn,tn |π 6�n,tn π′},

with Πn,tn housing continuous return-distribution vectors

We can apply Hausdorff’s metric to space Fn,tn of all nonemptyclosed subsets of Πn,tn ×Πn,tn :

dFn,tn (F1, F2) = inf(� > 0|F1 ⊆ (F2)� and F2 ⊆ (F1)�),

under which space Φn,tn of all (n, tn)-preferences is closed in Fn,tn

Our game is parameterized by member ϕ ≡ (ϕn,tn)n∈N,tn∈Tn ofΦ ≡

∏n∈N

∏tn∈Tn Φn,tn , a subset of F ≡

∏n∈N

∏tn∈Tn Fn,tn

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Continuity of Equilibrium Sets

Let us parameterize all equilibrium-related entities with ambiguityattitudes and their profiles—now Âan,tn(δ−n|ϕn,tn) is(n, tn)-player’s set of best-responding actions when opponentsmete out δ−n and his own attitude is ϕn,tn ; also, we have

B̂an,tn(δ−n|ϕn,tn), B̂dn,tn(δ−n|ϕn,tn), E

a(ϕ), and Ed(ϕ)

Âan,tn(·|·) is jointly upper hemi-continuous: suppose δk−n −→ δ−n,

ϕkn,tn −→ ϕn,tn , akn,tn −→ an,tn , and every

akn,tn ∈ Âan,tn(δ

k−n|ϕkn,tn), then an,tn ∈ Â

an,tn(δ−n|ϕn,tn); similar

results apply to B̂an,tn(·|·) and B̂dn,tn(·|·)

Finally, Ea(·) and Ed(·) are both upper hemi-continuous; inparticular, they are both closed at a fixed ϕ ≡ (ϕn,tn)n∈N,tn∈Tn

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Satisfaction and Alarmists’ Games

When preferences �n,tn are complete—either π 6�n,tn π′ orπ′ 6�n,tn π for any (π, π′)-pair, order-preserving utility functionsover return-distribution vectors can easily be identified

Suppose indeed each (n, tn)-player is associated with satisfactionfunction sn,tn so that π �n,tn π′ if and only if sn,tn(π) > sn,tn(π′)

When sn,tn ’s are continuous for satisfaction game, Ea 6= ∅; whenthey are further quasi-concave, Ed 6= ∅

For alarmists’ game where players assume the worst out of multiplepriors on state spaces Ωn,tn , satisfaction functions sn,tn are bothcontinuous and concave

This game has both action- and distribution-based equilibria

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The Enterprising Game

Though most attention was paid to ambiguity aversion as analternative ambiguity attitude, ambiguity seeking-ness could beequally prevalent; see, e.g., Charness, Karni, and Levin (2013)

We also believe that optimistic assessments of uncertain gains ispart of what drive people to participate in auctions, embark onexploratory journeys, and start new firms

Opposite to alarmists’ game, enterprising game uses satisfactionfunctions sn,tn(π) = supρ∈Pn,tns

0n,tn(π, ρ) where

s0n,tn(π, ρ) =

∫Rn,tn

un,tn(r) ·

[∫Ωn,tn

π(ω) · ρ(dω)

](dr)

This game has action-based equilibria: Ea 6= ∅24 / 37


Players for Alarmists’ and Enterprising Games

Who could be a player in an alarmists’ game?

To various degrees, most ordinary folks, a gambler who knowswhen to quit, a pension fund manager, or a doomsday prepper

Who could be a player in an enterprising game?

To various degrees, Alexander the Great, Hannibal Barca, ZhengHe, Christopher Columbus, Ferdinand Magellan, Captain JamesCook, Napoleon Bonaparte, Scarlet O’Hara, Charles Lindberg,Isoroku Yamamoto, George S. Patton, William (Bull) Halsey, anouveau riche art collector, Vito Corleone, Yuri A. Gagarin, Neil A.Armstrong, a shale gas field explorer, Steve Jobs, Bill Gates, ElonMusk, or a gambler who does not want to leave Vegas

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Risk- vs. Ambiguity-seeking

A gambler can decide whether or not to pick a ball from a box, andhe will win $1 when ball is red and lose $1 when ball is blue

If box is known to contain 50 red and 50 blue balls, then gamblerwill be considered risk-seeking if he wants to pick a ball

If numbers of red and blue balls in box are unknown, then gamblerwill be considered ambiguity-seeking if he wants to pick a ball

The gambler can be both risk-averse and ambiguity-seeking, for hemight lean more heavily towards “there being more than 50 redballs” than “there being fewer than 50 red balls”, and yet stillmight decide against picking a ball

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The Traditional Expected-utility Approach

In traditional approach, there are single priors ρn,tn on Ωn,tn andutility functions un,tn , so that every (n, tn)-player maximizes hisexpected utility over integrated return distribution

With conditional probabilities pn,tn|t−n and return-utility functionsvn,tn,t−n(an,tn , a−n,t−n) appropriately defined from primitives,

sn,tn(πan,tn(an,tn , δ−n)

)=∑

t−n∈T−n pn,tn|t−n××∫A−n,t−n

vn,tn,t−n(an,tn , a−n,t−n) ·∏m 6=n δm,tm(dam,tm)

and

sn,tn

(πdn,tn(δn,tn , δ−n)

)=

∫An,tn

sn,tn(πan,tn(a, δ−n)

)· δn,tn(da)

This certainly fits description of traditional expected-utility gameinvolving incomplete information

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Relations between Equilibrium Concepts

An important observation is

πdn,tn(δn,tn , δ−n) =

∫An,tn

πan,tn(a, δ−n) · δn,tn(da)

This prompts definitions of individual prominence and mixturepreservation for preference relations

When every �n,tn is individually prominent, one has Ed ⊆ Ea;when every �n,tn is mixture preserving, one has Ea ⊆ Ed

For satisfaction game, convexity of sn,tn will lead to individual

prominence of �n,tn , and hence Ed ⊆ Ea 6= ∅

Also, linearity of sn,tn will lead to both individual prominence and

mixture preservation of �n,tn , and hence Ed = Ea 6= ∅28 / 37


More Relations

With convex sn,tn ’s, enterprising game enjoys Ed ⊆ Ea 6= ∅

With linear sn,tn ’s, traditional game enjoys Ed = Ea 6= ∅

For general preference game, pure distribution-based equilibria are

always action-based ones: 1A ∩ Ed ⊆ 1A ∩ Ea

We can prove a higher-dimensional version of result that maximumof convex function over convex region comes from extreme points

Consequently, for enterprising game, pure distribution-basedequilibria are identical to their action-based counterparts:

1E ≡ 1A ∩ Ed = 1A ∩ Ea

For a more specialized enterprising game, we will show 1E 6= ∅

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Special Enterprising Game

All action spaces An,tn across different tn’s are the same An,which is in turn a finite set or compact real interval

Every Ωt is merely {t} × Ω̃ for some common Ω̃, which is productof finite sets or compact real intervals

Resulting return-utility function ũn,tn,t−n(an, a−n, ω̃) is increasingin ω̃ and has increasing differences between an and(tn, t−n, a−n, ω̃) as well as between (tn, t−n, a−n) and ω̃

With probability pn,tn|t−n player n believes unambiguously thatopponents’ type profile is at some t−n

Let distributions pn,tn ≡ (pn,tn|t−n)t−n∈T−n be increasing in tn inusual stochastic order

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More Strategic Complementarities

Player’s ambiguity on other factors is reflected by membership ofprior vector ν ≡ (νt−n)t−n∈T−n in subset Qn,tn of (P(Ω̃))T−n

Understand Qn,tn ⊆ (P(Ω̃))T−n as (∏tn∈T−n P̃n,tn,t−n)

⋂Kn,tn

Apply usual stochastic order to state distributions µ ∈ P(Ω̃), underwhich latter space is a lattice

Use induced set order on sublattices of P(Ω̃)

Let ambiguity sets P̃n,tn,t−n be sublattices of P(Ω̃) and alsoincreasing in tn in induced set order; let Kn,tn contain monotonemappings from T−n to P(Ω̃)

Focus on two special scenarios (A) and (B)

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Monotone Pure Equilibria

Suppose f has increasing differences between x and (y, z) as wellas between y and z, is supermodular in z, each Z̃(y) is asublattice, Z̃(·) is increasing in y, then with

g(x, y) = supz∈Z̃(y)

f(x, y, z),

function g will have increasing differences between x and y

This preservation of increasing differences might be compared toknown preservation of supermodularity stated in Topkis (1998)

Above, along with results in Milgrom and Roberts (1990), Milgromand Shannon (1994), Zhou (1994), and Yang and Qi (2013), leadto existence of monotone pure equilibria

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Monotone Comparative Statics

Suppose parameter λ ranges over partially ordered space Λ

Return-utility functions ũn,tn,t−n , distributions pn,tn , and ambiguity

sets P̃n,tn,t−n may all depend on λ ∈ Λ

Suppose dependencies are monotone in various senses

Then, we can establish monotone equilibria’s monotone movementwith parameter λ—a∗(λ) ≡ (a∗n,tn(λ))n∈N,tn∈Tn so that everya∗n,tn(λ) is increasing in both tn and λ

For scenario (A), our general message is consistent with one fortraditional expected-utility case in van Zandt and Vives (2007)

For scenario (B), our results can even be thought of asgeneralizations of existing ones

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Application to Auction

For auction involving n̄ bidders, we may let each type tn be signalthat bidder n receives, which is not necessarily his valuation ofitem being auctioned—think of offshore drilling right

An action set An,tn can be type-independent interval [0, w̄n], withw̄n being highest value bidder n assigns to item

Suppose ιn,tn,t−n(an,tn , a−n,t−n , ω) is 0-1 indicator of whetherbidder n wins item, υn,tn,t−n(ω) is bidder’s true valuation of item,and τn,tn,t−n(an,tn , a−n,t−n , ω) is his payment to seller

We can equate return rn,tn,t−n(an,tn , a−n,t−n , ω) toιn,tn,t−n(an,tn , a−n,t−n , ω) ·υn,tn,t−n(ω)−τn,tn,t−n(an,tn , a−n,t−n , ω)

34 / 37


Further Accommodations for Auction

Each state space Ωt can be Ψn̄ × {t} ×∏n̄n=1[0, w̄n], where Ψn̄ is

set of permutations on {1, ..., n̄} which is to play tie-breaking roles

A state ω ≡ (ψ, t, w) in Ωn,tn ≡⋃t−n∈T−n Ωtn,t−n can be written

as ((ψn)n=1,...,n̄, tn, (tm)m6=n, (wn)n=1,...,n̄), so bidder n with signaltn can have ambiguity on how ties are broken, opponents’ types,and everyone’s (including self’s) true valuation

Lo (1998) studied first- and second-price auctions where eachbidder knows his own valuation, possesses ambiguity on opponents’valuations, and uses worst-prior alarmists’ approach

It falls into our framework but for non-discrete types

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Summary of Results

Tracing of game definition to return-distribution vectors would giverise to preference game, more special satisfaction game, and evenmore special alarmists’ and enterprising games

Traditional game is within current framework which allows generalambiguity attitudes on external factors, and indirectly onopponents’ ambiguity attitudes and behaviors

Mild conditions can be identified for equilibrium existence andcontinuity of both action- and distribution-based types

Relations between the two equilibrium concepts are revealed

Monotone equilibrium trends are uncovered for special enterprisinggame involving strategic complementarities

Results can be applied to various real-life situations

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Future Research

This work has been published asYang, J. 2018. Game-theoretic Modeling of Players

Ambiguities on External Factors. Journal of MathematicalEconomics, 75, pp. 31-56

Future research possibilities still include:More general type spaces TnComputational schema of equilibriaAuction involving bidders ambiguous about item’s worths

Questions, Comments, Suggestions? Thank you.

37 / 37

Game-theoretic Modeling of Players' Ambiguities on External Factors

Documents

Game-theoretic Modeling of Players' Ambiguities on ......t n2T n;tn tn;t n (P(R n;tn)) n 14/37. Game-theoretic Modeling of Players’ Ambiguities on External Factors The Distribution-based