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Dynamic LP Games with Risk-Averse Players

Alejandro Toriello

Stewart School of Industrial and Systems EngineeringGeorgia Institute of Technology

IDS Seminar, College of Business AdministrationUniversity of Illinois, Chicago

September 15, 2017

joint with Nelson Uhan (USNA)

Sharing and Risk Aversion Don’t Mix:Dynamic LP Games with Risk-Averse Players

Alejandro Toriello

Stewart School of Industrial and Systems EngineeringGeorgia Institute of Technology

IDS Seminar, College of Business AdministrationUniversity of Illinois, Chicago

September 15, 2017

joint with Nelson Uhan (USNA)

MotivationCalifornia cut flower industry (Nguyen/Toriello/Dessouky/Moore 13)

I Study of potential consolidation center estimated up to 1/3reduction in transport costs. Issue deemed existential by somein industry.

I Results presented before U.S. Congress.

I Application submitted to USDOT for discretionary grant.

I However, many farmers skeptical of consolidation and wary ofpotential risks.

I Looming question: Can many disparate agents agree oncooperation and cost sharing? How does aversion to riskaffect the ability to cooperate?

MotivationCalifornia cut flower industry (Nguyen/Toriello/Dessouky/Moore 13)

I Study of potential consolidation center estimated up to 1/3reduction in transport costs. Issue deemed existential by somein industry.

I Results presented before U.S. Congress.

I Application submitted to USDOT for discretionary grant.

I However, many farmers skeptical of consolidation and wary ofpotential risks.

I Looming question: Can many disparate agents agree oncooperation and cost sharing? How does aversion to riskaffect the ability to cooperate?

Outline

IntroductionA Crash Course in Cooperative GamesPrimer on Coherent Risk Measures

Model FormulationThe Risk-Averse Strong Sequential CoreImplications on CooperationApplication: Risk-Averse Newsvendor Games

Parting Thoughts

Cost Allocation in OR Models(Static) cooperative games and the core

I Players: N = {1, . . . , n}, e.g. retailers or producers.

I Cost function: f : 2N → R, e.g. joint venture cost when somesubset of players cooperates.

I Goal: Assuming all players in N cooperate, find a costallocation χ ∈ RN that splits cost “fairly.”

I Usually models situations in which agents enter into bindingagreement.

I Especially when cooperation occurs over time.

Cost Allocation in OR Models(Static) cooperative games and the core

I Players: N = {1, . . . , n}, e.g. retailers or producers.

I Cost function: f : 2N → R, e.g. joint venture cost when somesubset of players cooperates.

I Goal: Assuming all players in N cooperate, find a costallocation χ ∈ RN that splits cost “fairly.”

I Usually models situations in which agents enter into bindingagreement.

I Especially when cooperation occurs over time.

Cost Allocation in OR ModelsCooperative games and the core

I Core (Gillies 59): Set of allocations χ that are

efficient:∑

N χi ≥ f(N),

stable: for U ⊆ N , no allocation ξ has∑U ξi ≥ f(U) and ξi < χi, i ∈ U.

No player or coalition can do better by defecting.

I Equivalently described by{χ ∈ RN :

∑Nχi ≥ f(N);

∑Uχi ≤ f(U), U ⊆ N

}.

I Non-empty core suggests cooperation is possible.

Cost Allocation in OR ModelsCooperative games and the core

I Core (Gillies 59): Set of allocations χ that are

efficient:∑

N χi ≥ f(N),

stable: for U ⊆ N , no allocation ξ has∑U ξi ≥ f(U) and ξi < χi, i ∈ U.

No player or coalition can do better by defecting.

I Equivalently described by{χ ∈ RN :

∑Nχi ≥ f(N);

∑Uχi ≤ f(U), U ⊆ N

}.

I Non-empty core suggests cooperation is possible.

Some Application ExamplesProduction, inventory and supply chain management

I Newsvendor and inventory centralization.I Chen (09), Chen/Zhang (09), Hartman/Dror (96,05),

Hartman/Dror/Shaked (00), Montrucchio/Scarsini (07),

Muller/Scarsini/Shaked (02), Ozen/Fransoo/Norde/Slikker (08),

Slikker/Fransoo/Wouters (05)

I Economic lot-sizingI Chen/Zhang (16), Gopaladesikan/Uhan/Zou (12), Toriello/Uhan

(14), van den Heuvel/Borm/Hamers (07)

I Inventory routingI Ozener/Ergun/Savelsbergh (13)

I Joint replenishmentI He/Zhang/Zhang (12), Zhang (09)

Linear Production GamesOwen (75)

f(U) = minx≥0

cx

= maxλ

λ(∑

Udi)

s.t. Ax =∑

U di

s.t. λA ≤ c

TheoremIf λ is dual optimal for f(N),

χi = λdi, i ∈ N

is in the core of f .

Proof.strong duality⇒ efficiency weak duality⇒ stability

Linear Production GamesOwen (75)

f(U) = minx≥0

cx

= maxλ

λ(∑

Udi)

s.t. Ax =∑

U di

s.t. λA ≤ c

TheoremIf λ is dual optimal for f(N),

χi = λdi, i ∈ N

is in the core of f .

Proof.strong duality⇒ efficiency weak duality⇒ stability

Linear Production GamesOwen (75)

f(U) = minx≥0

cx = maxλ

λ(∑

Udi)

s.t. Ax =∑

U di s.t. λA ≤ c

TheoremIf λ is dual optimal for f(N),

χi = λdi, i ∈ N

is in the core of f .

Proof.strong duality⇒ efficiency weak duality⇒ stability

Cooperation under UncertaintyDynamic LP games

I Players face two-stage decision process defined by

f1(U) := minx,s≥0

c1x1 + h1s1 + E[cτxτ + hτsτ ]

s.t. A1x1 − C1s1 =∑

Ud1i (stage 1)

Atxt +Bts1 − Ctst =∑

Udti, t ∈ D, (stage 2)

x’s are actions (e.g. orders) and s’s are states (e.g. inventory).

I Static core does not capture timing, need dynamic allocation.I χt

i: What player i pays if t realizes.

Cooperation under UncertaintyDynamic LP games

I Players face two-stage decision process defined by

f1(U) := minx,s≥0

c1x1 + h1s1 + E[cτxτ + hτsτ ]

s.t. A1x1 − C1s1 =∑

Ud1i (stage 1)

Atxt +Bts1 − Ctst =∑

Udti, t ∈ D, (stage 2)

x’s are actions (e.g. orders) and s’s are states (e.g. inventory).

I Static core does not capture timing, need dynamic allocation.I χt

i: What player i pays if t realizes.

Cooperation under UncertaintyDynamic LP games

I Suppose players implement optimal solution x, s.

I Strong sequential core (Predtetchinski/Herings/Peters 02):Allocations χ that are

stage-wise efficient: pay as you go,∑Nχ

ti ≥ ctxt + htst, t ∈ D ∪ 1,

time-consistently stable: never vulnerable to any U defecting,∑U

(χ1i + E[χτi ]

)≤ f1(U) (stage 1)∑

Uχti ≤ f

t(U, s1U ), t ∈ D (stage 2)

where f t is static problem faced by U in scenario t.

Cooperation under UncertaintyDynamic LP games

I Suppose players implement optimal solution x, s.

I Strong sequential core (Predtetchinski/Herings/Peters 02):Allocations χ that are

stage-wise efficient: pay as you go,∑Nχ

ti ≥ ctxt + htst, t ∈ D ∪ 1,

time-consistently stable: never vulnerable to any U defecting,∑U

(χ1i + E[χτi ]

)≤ f1(U) (stage 1)∑

Uχti ≤ f

t(U, s1U ), t ∈ D (stage 2)

where f t is static problem faced by U in scenario t.

Cooperation under UncertaintyDynamic LP games

I Suppose players implement optimal solution x, s.

I Strong sequential core (Predtetchinski/Herings/Peters 02):Allocations χ that are

stage-wise efficient: pay as you go,∑Nχ

ti ≥ ctxt + htst, t ∈ D ∪ 1,

time-consistently stable: never vulnerable to any U defecting,∑U

(χ1i + E[χτi ]

)≤ f1(U) (stage 1)∑

Uχti ≤ f

t(U, s1U ), t ∈ D (stage 2)

where f t is static problem faced by U in scenario t.

Cooperation under UncertaintyDynamic LP games

Theorem (e.g. Xu/Veinott 13)

A dual-based dynamic allocation similar to Owen’s is in the strongsequential core.

I Similar results extend to multi-stage models.

Example: Newsvendor GamesInventory pooling

I Each player faces uncertain demand dti, can order now,backlog or salvage later:

f1nv(U) := minx,s≥0

x1 + E[bxτ − vsτ ]

s.t. x1 + xt − st =∑

Udti, t ∈ D.

I Suppose scenarios ordered by∑

N dti. Then critical index is

t := maxt∈D

P(demand ≥∑

Ndti) ≥ 1−v

b−v .

Optimal order is x1 =∑

N dti.

Example: Newsvendor GamesInventory pooling

I Each player faces uncertain demand dti, can order now,backlog or salvage later:

f1nv(U) := minx,s≥0

x1 + E[bxτ − vsτ ]

s.t. x1 + xt − st =∑

Udti, t ∈ D.

I Suppose scenarios ordered by∑

N dti. Then critical index is

t := maxt∈D

P(demand ≥∑

Ndti) ≥ 1−v

b−v .

Optimal order is x1 =∑

N dti.

Example: Newsvendor GamesInventory pooling

I If s1i = dti, i.e. players split order according to demand in t,

χ1i = dti, χti =

b(dti − dti), t > t

−v(dti − dti), t < t

0, t = t

is in the SSC.

I Each player pays to order their demand under scenario t.

I Second-stage order/salvage and side payments occur at priceb/v depending on realization.

Coherent Risk MeasuresArtzner/Delbaen/Eber/Heath (99)

I We now suppose players may be risk-averse.

I Assume risk preferences captured by coherent risk measure,ρ : RD → R with

monotonicity: ρ(χτ ) ≤ ρ(ξτ ) if χ ≤ ξ,

translation invariance: ρ(χτ + κ) = ρ(χτ ) + κ for constant κ,

positive homogeneity: ρ(κχτ ) = κρ(χτ ) for constant κ ≥ 0,

subadditivity: ρ(χτ + ξτ ) ≤ ρ(χτ ) + ρ(ξτ ) (convexity)

I Expectation is risk-neutral, additive instead of subadditive.

Coherent Risk MeasuresArtzner/Delbaen/Eber/Heath (99)

I We now suppose players may be risk-averse.

I Assume risk preferences captured by coherent risk measure,ρ : RD → R with

monotonicity: ρ(χτ ) ≤ ρ(ξτ ) if χ ≤ ξ,

translation invariance: ρ(χτ + κ) = ρ(χτ ) + κ for constant κ,

positive homogeneity: ρ(κχτ ) = κρ(χτ ) for constant κ ≥ 0,

subadditivity: ρ(χτ + ξτ ) ≤ ρ(χτ ) + ρ(ξτ ) (convexity)

I Expectation is risk-neutral, additive instead of subadditive.

Coherent Risk MeasuresArtzner/Delbaen/Eber/Heath (99)

I We now suppose players may be risk-averse.

I Assume risk preferences captured by coherent risk measure,ρ : RD → R with

monotonicity: ρ(χτ ) ≤ ρ(ξτ ) if χ ≤ ξ,

translation invariance: ρ(χτ + κ) = ρ(χτ ) + κ for constant κ,

positive homogeneity: ρ(κχτ ) = κρ(χτ ) for constant κ ≥ 0,

subadditivity: ρ(χτ + ξτ ) ≤ ρ(χτ ) + ρ(ξτ ) (convexity)

I Expectation is risk-neutral, additive instead of subadditive.

Coherent Risk MeasuresArtzner/Delbaen/Eber/Heath (99)

I We now suppose players may be risk-averse.

I Assume risk preferences captured by coherent risk measure,ρ : RD → R with

monotonicity: ρ(χτ ) ≤ ρ(ξτ ) if χ ≤ ξ,

translation invariance: ρ(χτ + κ) = ρ(χτ ) + κ for constant κ,

positive homogeneity: ρ(κχτ ) = κρ(χτ ) for constant κ ≥ 0,

subadditivity: ρ(χτ + ξτ ) ≤ ρ(χτ ) + ρ(ξτ ) (convexity)

I Expectation is risk-neutral, additive instead of subadditive.

Coherent Risk MeasuresArtzner/Delbaen/Eber/Heath (99)

I We now suppose players may be risk-averse.

I Assume risk preferences captured by coherent risk measure,ρ : RD → R with

monotonicity: ρ(χτ ) ≤ ρ(ξτ ) if χ ≤ ξ,

translation invariance: ρ(χτ + κ) = ρ(χτ ) + κ for constant κ,

positive homogeneity: ρ(κχτ ) = κρ(χτ ) for constant κ ≥ 0,

subadditivity: ρ(χτ + ξτ ) ≤ ρ(χτ ) + ρ(ξτ ) (convexity)

I Expectation is risk-neutral, additive instead of subadditive.

Coherent Risk MeasuresArtzner/Delbaen/Eber/Heath (99)

TheoremAny coherent risk measure has a robust representation as aworst-case expectation over a closed, convex set of distributions,

ρ(χτ ) = maxq∈Q

Eq[χτ ]

for Q ⊆ ∆D.

I Our results extend to multi-stage models using conditionalrisk mappings (Ruszczynski/Shapiro 06).

Including Risk Aversion in LP Game

I Suppose players face same situation.

I Linear dynamics, additive demand/requirements, two stages.

I But each player i assesses uncertain costs based on coherentrisk measure ρi.

I What solution should they implement?

I Given a solution, how are costs split? What do players think isfair?

The Risk-Averse Strong Sequential Core

If the players implement solution x, s, the risk-averse SSC hasallocations χ with

stage-wise efficiency:∑Nχ

ti ≥ ctxt + htst, t ∈ D ∪ 1,

time-consistent stability: for coalition U ⊆ N ,

stage 1 No other solution x′, s′ satisfies U ’s demandand has a stage-wise efficient allocation ξ with

ξ1i + ρi(ξτi ) < χ1

i + ρi(χτi ), i ∈ U,

stage 2∑

Uχti ≤ f t(U, s1U ), t ∈ D.

The Risk-Averse Strong Sequential Core

If the players implement solution x, s, the risk-averse SSC hasallocations χ with

stage-wise efficiency:∑Nχ

ti ≥ ctxt + htst, t ∈ D ∪ 1,

time-consistent stability: for coalition U ⊆ N ,

stage 1 No other solution x′, s′ satisfies U ’s demandand has a stage-wise efficient allocation ξ with

ξ1i + ρi(ξτi ) < χ1

i + ρi(χτi ), i ∈ U,

stage 2∑

Uχti ≤ f t(U, s1U ), t ∈ D.

The Risk-Averse Strong Sequential Core

If the players implement solution x, s, the risk-averse SSC hasallocations χ with

stage-wise efficiency:∑Nχ

ti ≥ ctxt + htst, t ∈ D ∪ 1,

time-consistent stability: for coalition U ⊆ N ,

stage 1 No other solution x′, s′ satisfies U ’s demandand has a stage-wise efficient allocation ξ with

ξ1i + ρi(ξτi ) < χ1

i + ρi(χτi ), i ∈ U,

stage 2∑

Uχti ≤ f t(U, s1U ), t ∈ D.

The Risk-Averse Strong Sequential Core

TheoremThe SSC for solution x, s is the set of allocations χ with∑

Nχti ≥ ctxt + htst, t ∈ D ∪ 1, (efficiency)∑

U

(χ1i + ρi(χ

τi ))≤ f1(U), U ⊆ N (stage-1 stable)∑

Uχti ≤ f t(U, s1U ), U ⊆ N, t ∈ D, (stage-2 stable)

where

f1(U) := minx,s≥0;ξ

∑U

(ξ1i + ρi(ξ

τi ))

s.t. A1x1 − C1s1 =∑

Ud1i

Atxt +Bts1 − Ctst =∑

Udti, t ∈ D,∑

Uξti ≥ ctxt + htst, t ∈ D ∪ 1.

The Risk-Averse Strong Sequential Core

TheoremThe SSC for solution x, s is the set of allocations χ with∑

Nχti ≥ ctxt + htst, t ∈ D ∪ 1, (efficiency)∑

U

(χ1i + ρi(χ

τi ))≤ f1(U), U ⊆ N (stage-1 stable)∑

Uχti ≤ f t(U, s1U ), U ⊆ N, t ∈ D, (stage-2 stable)

where

f1(U) := minx,s≥0;ξ

∑U

(ξ1i + ρi(ξ

τi ))

s.t. A1x1 − C1s1 =∑

Ud1i

Atxt +Bts1 − Ctst =∑

Udti, t ∈ D,∑

Uξti ≥ ctxt + htst, t ∈ D ∪ 1.

The Risk-Averse Strong Sequential CoreStructural implications

If the players implement x, s and use allocation χ in the SSC,

risk optimality: x, s and χ are optimal for f1(N),∑N

(χ1i + ρi(χ

τi ))

= f1(N),

convexity: if the risk measures ρi are convex, so is the SSC.

Consequences for CooperationRisk alignment

Corollary

If χ is in the SSC, ∑Nρi(χ

τi ) =

∑NEq[χτi ],

for some q ∈ ∆D.

I The allocation must be risk-aligned; the same distributionmust yield worst-case expected cost for all players.

I E.g. player A cannot hope for low demand if player B hopesfor high demand.

Consequences for CooperationRisk alignment

Corollary

If χ is in the SSC, ∑Nρi(χ

τi ) =

∑NEq[χτi ],

for some q ∈ ∆D.

I The allocation must be risk-aligned; the same distributionmust yield worst-case expected cost for all players.

I E.g. player A cannot hope for low demand if player B hopesfor high demand.

Consequences for CooperationPooling versus stability

Corollary

If player j is equally or less risk-averse than all others,

Qj ⊆ Qi, i ∈ N \ j,

then f1 can be optimized by assigning all cost to j.

I Optimization requires the pooling of cost (subadditivity),stability requires this cost spread out without increasing risk.

I May only be possible by assigning all uncertainty to j, whileother players get scenario-independent side payments.

Consequences for CooperationPooling versus stability

Corollary

If player j is equally or less risk-averse than all others,

Qj ⊆ Qi, i ∈ N \ j,

then f1 can be optimized by assigning all cost to j.

I Optimization requires the pooling of cost (subadditivity),stability requires this cost spread out without increasing risk.

I May only be possible by assigning all uncertainty to j, whileother players get scenario-independent side payments.

Consequences for CooperationPooling versus stability

Corollary

If player j is equally or less risk-averse than all others,

Qj ⊆ Qi, i ∈ N \ j,

then f1 can be optimized by assigning all cost to j.

I Optimization requires the pooling of cost (subadditivity),stability requires this cost spread out without increasing risk.

I May only be possible by assigning all uncertainty to j, whileother players get scenario-independent side payments.

Consequences for CooperationCompatible beliefs

Corollary

The SSC is empty if ⋂NQi = ∅.

I If players cannot agree on a common belief (distribution)about the future, they cannot cooperate.

I E.g. player A believes tomorrow will be sunny, B believes itwill be cloudy. Each can bet arbitrary amount on their beliefto create artificial risk arbitrage.

Consequences for CooperationCompatible beliefs

Corollary

The SSC is empty if ⋂NQi = ∅.

I If players cannot agree on a common belief (distribution)about the future, they cannot cooperate.

I E.g. player A believes tomorrow will be sunny, B believes itwill be cloudy. Each can bet arbitrary amount on their beliefto create artificial risk arbitrage.

Risk-Averse Newsvendor Games

I Assume players assess risk with same comonotonic measure, ρ.

f1nv(U) := minx,s≥0

x1 + ρ(bxτ − vsτ )

s.t. x1 + xt − st =∑

Udti, t ∈ D.

comonotonicity: ρ(χτ + ξτ ) = ρ(χτ ) + ρ(ξτ ) if χ, ξ comonotonic

I Critical scenario index t defined with worst-case distribution.

I Still optimal to order x1 =∑

N dti in first stage.

Risk-Averse Newsvendor Games

I Assume players assess risk with same comonotonic measure, ρ.

f1nv(U) := minx,s≥0

x1 + ρ(bxτ − vsτ )

s.t. x1 + xt − st =∑

Udti, t ∈ D.

comonotonicity: ρ(χτ + ξτ ) = ρ(χτ ) + ρ(ξτ ) if χ, ξ comonotonic

I Critical scenario index t defined with worst-case distribution.

I Still optimal to order x1 =∑

N dti in first stage.

Risk-Averse Newsvendor Games(Not so) good news

I Recall solution and allocation with s1i = dti and

χ1i = dti, χti =

b(dti − dti), t > t

−v(dti − dti), t < t

0, t = t.

TheoremSuppose players’ demand is comonotonic, d1i ≤ d2i ≤ . . . for i ∈ N .Then χ is in the SSC.

I If demand is comonotonic, players perceive least benefit fromcooperation.

I χ mirrors what each player would incur on their own.

Risk-Averse Newsvendor Games(Not so) good news

I Recall solution and allocation with s1i = dti and

χ1i = dti, χti =

b(dti − dti), t > t

−v(dti − dti), t < t

0, t = t.

TheoremSuppose players’ demand is comonotonic, d1i ≤ d2i ≤ . . . for i ∈ N .Then χ is in the SSC.

I If demand is comonotonic, players perceive least benefit fromcooperation.

I χ mirrors what each player would incur on their own.

Risk-Averse Newsvendor GamesBad news: an example with empty SSC

I Two players, exactly complementary demand,

d011 = d102 = 0, d101 = d012 = 1.

Ideal situation: Optimal joint order is 1, no uncertainty.

I Risk measure defined by Q = {(q, 1− q), (1− q, q)}.

I If v > 0, b big enough, q 6= 1/2, the SSC is empty.

I No matter how players split the order, the SSC requires the“losing” player to receive v per unit, but the “winning” playerto not pay anything.

I Holds for any q 6= 1/2, any risk measure that isn’t risk-neutral.

Risk-Averse Newsvendor GamesBad news: an example with empty SSC

I Two players, exactly complementary demand,

d011 = d102 = 0, d101 = d012 = 1.

Ideal situation: Optimal joint order is 1, no uncertainty.

I Risk measure defined by Q = {(q, 1− q), (1− q, q)}.

I If v > 0, b big enough, q 6= 1/2, the SSC is empty.

I No matter how players split the order, the SSC requires the“losing” player to receive v per unit, but the “winning” playerto not pay anything.

I Holds for any q 6= 1/2, any risk measure that isn’t risk-neutral.

Risk-Averse Newsvendor GamesBad news: an example with empty SSC

I Two players, exactly complementary demand,

d011 = d102 = 0, d101 = d012 = 1.

Ideal situation: Optimal joint order is 1, no uncertainty.

I Risk measure defined by Q = {(q, 1− q), (1− q, q)}.

I If v > 0, b big enough, q 6= 1/2, the SSC is empty.

I No matter how players split the order, the SSC requires the“losing” player to receive v per unit, but the “winning” playerto not pay anything.

I Holds for any q 6= 1/2, any risk measure that isn’t risk-neutral.

Conclusions

I In dynamic LP games, cooperation always possible whenplayers are risk neutral.

I When players are risk-averse, cooperation much more difficult.

I SSC can easily be empty, implying cooperation unlikely.

I We can only guarantee SSC is non-empty when cooperation isleast beneficial.

I Results suggest risk neutrality is important necessarycondition in cooperation.

I Risk aversion may explain lack of cooperation in situationswhere risk-neutral models predict it.