Existence and Characteristics of the asymptotic Nash Equilibrium . 113

PART III Toward Efficiency: Secondary Market 100

VIII.2 Large Markets: Asymptotic Analysis

VIII.2.3 Existence and Characteristics of the asymptotic Nash Equilibrium . 113

Combining the three cases of the realization ofQ, we conclude:

EQ−i

P(Q,x)|_(Qi=q0)

−EQ−i

P(Q,x)|_(Qi=q00) =O

q_max min(σ√

n,π(n))

(VIII.21)

Together Theorem VIII.2.1 and VIII.2.2, we conclude that an individual’s influence on the price in the secondary market is O_max(x

max,q_max) min(σ√

n,π(n))

as n→+∞. In other words, for playeri∈N, for anyx_i,x_−iandQ_i, we have:

EQ−i

P(Q,x) Q_i

=EQ−i[P(Q_−i,x_−i)]±O

max(x_max,q_max) min(σ√

n,π(n))

. (VIII.22)

x_max, we have EQ

min(P(Q,x),c⁰_i)|(x_i=x⁰)

−EQ

min(P(Q,x),c⁰_i)|(x_i=x⁰⁰)

=O

x_max min(σ√

n,π(n))

(VIII.23)

as n→+∞, given{x₁, . . . ,x_i−1,x_i+1, . . . ,x_n}have been determined.

Proof (Sketch). Based on Theorem VIII.2.1,

EQ

P(Q,x)|(x_i=x⁰)

−EQ

P(Q,x)|(x_i=x⁰⁰) =O

x_max min(σ√

n,π(n))

(VIII.24)

As

EQ[c⁰_i|(x_i=x⁰)]−EQ[c⁰_i|(x_i=x⁰⁰)]

=0, we coud get the result.

Lemma VIII.2.3. For player i∈N, assume that {Q_−i} are independent and identically distributed with finite mean µ and varianceσ². For any q⁰and q⁰⁰ s.t. 0≤q⁰,q⁰⁰ ≤q_max, we have

EQ_−i

min(P(Q,x),c⁰_i)|(Q_i=q⁰)

−EQ_−i

min(P(Q,x),c⁰_i)|(Q_i=q⁰⁰)

=O

q_max min(σ√

n,π(n))

(VIII.25)

as n→+∞, given{x₁,x₂. . . ,x_n}have been determined.

Proof (Sketch). Based on Theorem VIII.2.2,

EQ−i

P(Q,x)|(Q_i=q⁰)

−EQ−i

P(Q,x)|(Q_i=q⁰⁰) =O

q_max min(σ√

n,π(n))

(VIII.26) As

EQ−i[c⁰_i|(Q_i=q⁰)]−EQ−i[c⁰_i|(Q_i=q⁰⁰)]

=0, we could get the result.

Intuitively, the two lemmas show that player i’s influence to min(P(Q,x),c⁰_i) is also O

1 min(√

n,π(n))

asn→+∞. Similarly as Equation VIII.22, for playeri∈N, for anyx_i,x_−i

andQ_i, we have

EQ−i

min(P(Q,x),c⁰_i) Q_i

=EQ−i

min(P(Q_−i,x_−i),c⁰_i)

±O

max(x_max,q_max) min(σ√

n,π(n))

(VIII.27) We now reorganize the (expected) utility function in Equation VIII.7. For easiness of notation, letψ(n) = ^max(x_min(σ^max√_n,π(n))^,qmax), andP=P(Q_−i,x_−i)in the following discussion.

U_i(x_i,x_−i)

=EQ

−cx_i+ (x_i−Q_i)⁺P(Q,x)−(Q_i−x_i)⁺min P(Q,x),c⁰_i

=EQi

n

−cx_i+ (x_i−Q_i)⁺EQ−i

P(Q,x) Q_i

−(Q_i−x_i)⁺EQ−i

min P(Q,x),c⁰_i Q_io

=EQi

n−cx_i+ (x_i−Q_i)⁺EQ_−i[P(Q_−i,x_−i)]

−(Q_i−x_i)⁺EQ−i

min P(Q_−i,x_−i),c⁰_io

±O(ψ(n))

=EQi

EQ−i(P)−c

x_i+ (Q_i−x_i)⁺

EQ−i(P))−EQ−i

min P,c⁰_i

−Q_iEQ−i(P) ±O(ψ(n))

(VIII.28)

It is easy to see thatEQ−i(P)−EQ−i[min(P,c⁰_i)]≥0, andU_i(x_i,x_−i)is convex forx_i, asn→∞. Then the best response of playerican be got only whenx_i=0 orx_i=x_max.

• Whenx_i=0, thenU_i(x_i,x_−i) =−EQi(Q_i)EQ−i[min(P,c⁰_i)]±O(ψ(n));

• Whenx_i=x_max, thenU_i(x_i,x_−i) =

EQ−i(P)−c

x_max−EQi(Q_i)EQ−i(P)±O(ψ(n)).

Let A=EQi(Q_i)

EQ−i(P)−EQ−i[min(P,c⁰_i)] , and B=

EQ−i(P)−c

x_max, then the best response of playeri(asn→∞) is

x^∗_i =



















0, A>B

0 orx_max, A=B x_max, A<B

(VIII.29)

Note that whenA=B, playeriis indifferent between 0 andx_max.

We now discuss the existence and characteristics of asymptotic Nash equilibrium among players, as shown in Theorem VIII.2.3.

Theorem VIII.2.3. Asymptotic Nash equilibrium always exists, andx^∗={x^∗₁,x^∗₂, . . . ,x^∗_n} is an asymptotic Nash equilibriumif and only ifthe following conditions holds,

1. For any player i∈N, there exists a thresholdθ^∗, in which

x^∗_i =



















0, c⁰_i<θ^∗ 0or x_max, c⁰_i=θ^∗ x_max, c⁰_i>θ^∗

(VIII.30)

2. EQ[P(Q,x^∗)] =c as n→+∞.

Proof. Firstly we prove that if a strategy profilex^∗satisfies the two conditions above, then it is an asymptotic Nash equilibrium. Assume that there is a thresholdθ^∗, such that players inN₊ ={i∈N|c⁰_i>θ^∗}orderx_max and players inN₋ ={i∈N|c⁰_i<θ^∗}order 0, and for correspondingx^∗,EQ[P(Q,x^∗)] =cholds asn→∞, we now show thatx^∗is an asymptotic Nash equilibrium.

When EQ[P(Q,x^∗)] =c, as EQ[P(Q,x^∗)] =EQ−i

P(Q_−i,x^∗_−i)

±O(ψ(n)), then we could get EQ[P(Q,x^∗)] =EQ_−i

P(Q_−i,x^∗_−i)

as n→+∞. And Equation VIII.28 can be simplified as follows (letP=P(Q_−i,x^∗_−i)),

U_i(x^∗_i,x^∗_−i)

=EQi

(Q_i−x^∗_i)⁺

EQ−i(P)−EQ−i

min(P,c⁰_i) −Q_ic ±O(ψ(n))

(VIII.31)

AsEQ−i(P)−EQ−i[min(P,c⁰_i)]≥0, we could get thatU_i(x^∗_i,x^∗_−i)is anon-increasing function withx^∗_i (asn→∞).

• For any player i that in N₋ = {i∈N|c⁰_i <θ^∗}, her best response is x^∗_i =0. As U_i(x^∗_i,x^∗_−i)isnon-increasing, her expected utility cannot be improved.

• For any playerithat inN₊={i∈N|c⁰_i>θ^∗}, her best response isx^∗_i =x_max. We have known thatc⁰_i>θ^∗>P(based on Equation VIII.6), soEQ_−i(P)−EQ_−i[min(P,c⁰_i)] = 0. So we could get

U_i(x^∗_i,x^∗_−i) =−EQi(Q_i)c±O(ψ(n)) (VIII.32) Then playericannot gain by deviating.

Sox^∗is an asymptotic Nash equilibrium.

Then we show that if x^∗ is an asymptotic Nash equilibrium, then condition 1 holds.

It is easy to see that all players order 0 or all players orderx_max cannot be an asymptotic Nash equilibrium. Otherwise, all players will become buyers (or sellers) in the secondary market, and some players can deviate from their strategies to gain profit.

From Equation VIII.29, we observe that: Ifx^∗_i =x_max, then for any player jwithc⁰_j>c⁰_i, x^∗_j =x_max; Ifx_i^∗=0, then for any player j withc⁰_j<c⁰_i, x^∗_j =0. Then we could know that the condition 1 holds onx^∗.

Then we show that if strategy profile x^∗ is an asymptotic Nash equilibrium, then we could getEQ

P(Q,x^∗)

=casn→+∞. We prove it by contradiction:

• If EQ[P(Q,x^∗)]<c, we claim that players in N₊ ={i∈N|c⁰_i >θ^∗} (i.e. players those orderx_max) have incentive to deviate. WhenEQ[P(Q,x^∗)]<c, for playeriin N₊ ={i∈N|c⁰_i>θ^∗}, we haveEQ[P(Q,x^∗)] =EQ−i(P)asn→+∞, and

U_i(x^∗_i,x^∗_−i) =EQi

EQ−i(P)−c

x^∗_i −Q_iEQ−i(P) ±O(ψ(n)) (VIII.33)

AsEQ−i(P)−c<0, any playeri∈N₊has incentive to decrease thex_i. So it cannot be an asymptotic Nash equilibrium.

• IfEQ[P(Q,x^∗)]>c, we claim that there exists playeri∈N₋={i∈N|c⁰_i<θ^∗}(i.e.

players those order 0 unit of resources) has incentive to deviate. Intuitively, playeri withc⁰_ithat is very closed toθ^∗may have incentive to deviate. Based on Assumption VIII.2.1 and definition of market clearing price, for any smallδ, there existsε such that for playeriwithc⁰_i=θ^∗−ε, andc⁰_i≥P+δ. So we could get

U_i(x_i=0,x^∗_−i)

=EQi(Q_i)

EQ−i(P)−EQ−i(min(P,c⁰_i))

−EQi(Q_i)EQ−i(P)±O(ψ(n))

=0−EQi(Q_i)EQ−i(P)±O(ψ(n))

(VIII.34) IfEQ_−i(P)−c>0 asn→∞, then

U_i(x_i=x_max,x^∗_−i)

=EQi(Q_i)

EQ−i(P)−c

x_max−EQi(Q_i)EQ−i(P)±O(ψ(n))

>0−EQi(Q_i)EQ−i(P)±O(ψ(n))

=U_i(x_i=0,x^∗_−i)

(VIII.35)

Then playerihas incentive to deviate andx^∗ cannot be an asymptotic Nash equilib- rium.

So we conclude thatEQ[P(Q,x^∗)] =casn→+∞.

Recall that we have assumed that x_max ≥q_max. If all players order either 0 or x_max in the primary market, then we can always adjust the proportion of players those order 0 orx_max to make EQ[P(Q,x^∗)] =c hold as n→∞. So the existence of the equilibrium is straightforward.

To sum up, asymptotic Nash equilibrium exists and x^∗ is an asymptotic Nash equilib- riumif and only if the two conditions hold.

Intuitively, Theorem VIII.2.3 warrants the existence of the asymptotic Nash equilib- rium and characterizes the equilibrium. In the equilibrium, players are divided into two

sets based on the marginal penalty of shortage c⁰. Those players with relatively high c⁰ would order as many resources as they can in the primary market and become sellers in the secondary market, and players with relatively low c⁰ would not order resources and wait to buy resources in the secondary market. More importantly, in equilibrium, the expected price in the secondary market is equal to the price in the primary market, as the number of players approaches infinity. The properties of the asymptotic equilibrium imply that secondary market could mitigate the demand uncertainty in the primary market.

VIII.2.4 Social Welfare and Aggregated Orders

Based on the existence and characteristics of asymptotic Nash equilibrium in Theorem VIII.2.3, we discuss the social welfare (i.e. ∑_i∈NU(x_i,x_−i)) and the aggregated orders (i.e. ∑_i∈Nx_i) in the game with a secondary market, comparing these to the social welfare and aggregated orderswithout a secondary market. We find that both social welfare and aggregated orderswitha secondary market are significantly better than thosewithoutwhen the number of players is approaching infinity.

We begin by discussing theoptimalsocial welfare and aggregated orders. Theoptimal social welfare and aggregated orders in this case can be obtained when we treat all players as asingleplayer, in which the (stochastic) demand is∑_i∈NQ_i. Assume players’ demands are independent and identically distributed, letQ denote the random demand of a player,

andq₁,q₂, . . . ,q_n be a sequence of realization of random variableQ. Based on the Law of

Large Numbers, asn→+∞, ¹_n∑i∈Nq_i→EQ(Q). Consequently, the authority can simply ordernEQ(Q)resources and distribute resources to players according to their needs at price c. LetSW_#andOD_#denote theoptimalsocial welfare and aggregated orders, respectively.

Then we can easily obtain that

SW_#=−cnEQ(Q) (VIII.36)

and

OD_#=nEQ(Q). (VIII.37)

LetSW_2nd denote the social welfare in the casewithouta secondary market, where each player makes an independent choice about the optimal amount of resource to order (note that this is also the optimal social welfare in the setting without a secondary market, since all player decisions are entirely decoupled). Correspondingly, letSW_2nd denote the social welfarein the equilibriumof the gamewitha secondary market. Similarly, letOD_2nd and OD_2nd denote the aggregated order of players in cases of without and with a secondary market (in the latter case, in equilibrium), respectively.

Our main results in this section (Theorems VIII.2.4 and VIII.2.5) show that the differ- ence between the optimal and equilibrium outcomes is bounded by O(max(√

n, ⁿ

π(n))), a significant improvement over theΘ(n) difference between the optimum and the outcome with only the newsvendor model (i.e., without the secondary market).

Theorem VIII.2.4. The difference between optimal social welfare and the social welfare withoutsecondary market isΘ(n), i.e.

SW_#−SW_2nd

=Θ(n), as n→+∞.

Proof. Based on Equation VIII.2, without secondary market, for player i, the (expected) utility isU_i(x^∗_i) =EQ[−cx^∗_i −c⁰_i(Q−x^∗_i)⁺], in whichx^∗_i =F_Q⁻¹_c0

i−c c⁰

. And it can be also denoted as U_i(x^∗_i) = −cEQ(Q)±Θ(1). As players are independent, the social welfare is SW_2nd =∑_i∈NU_i(x^∗_i) =−cnEQ(Q)±Θ(n) as n→+∞. So

SW_#−SW_2nd

=Θ(n) as n→+∞.

Theorem VIII.2.5. The difference between optimal social welfare and the social welfare in the equilibrium with secondary market is upper bounded by O(n·ψ(n)), i.e.

SW_#− SW_2nd

=O(n·ψ(n))as n→+∞, whereψ(n) = _min(σ^max(xmax√_n,π^,qmax_(n))⁾.

Proof. With secondary market, based on Theorem VIII.2.3, in equilibrium,EQ(P(Q,x)) = casn→+∞. For any playeri,EQ−i[P(Q_−i,x_−i)] =c±O(ψ(n)).

We have known that in equilibrium, players’s strategy is either 0 orx_max,

• Ifx_i=0, then

U_i(x_i=0,x_−i) =−EQi

Q_iEQ_−i

min(P(Q_−i,x_−i),c⁰_i) ±O(ψ(n))

=−EQi(Q_i)EQ−i

min(P(Q_−i,x_−i),c⁰_i)

±O(ψ(n))

≥ −EQi(Q_i)EQ−i[P(Q_−i,x_−i)]±O(ψ(n))

=−EQi(Q_i)c±O(ψ(n))

(VIII.38)

• Ifx_i=x_max,

U_i(x_i=x_max,x_−i) =EQ_i

−cx_max+ (x_max−Q_i)EQ_−i[P(Q_−i,x_−i)]

±O(ψ(n))

=EQi[−cx_max+ (x_max−Q_i)c]±O(ψ(n))

=−EQ_i(Q_i)c±O(ψ(n))

(VIII.39)

To sum them up, then (ignoring the constant factor):

SW_2nd =

∑

i∈N

U_i(x_i,x_−i)≥ −cnEQ(Q)±O(n·ψ(n)) (VIII.40)

So we could get that:

|SW_#−SW_2nd|=O(n·ψ(n)) (VIII.41)

Similarly, the sum of orders in the equilibrium of the gamewitha secondary market is also much closer to the optimal sum of orders than the casewithouta secondary market.

Theorem VIII.2.6. The difference between the aggregated orderswithoutsecondary mar- ket and the optimal aggregated orders isΘ(n), i.e.

OD_2nd−OD_#

=Θ(n), as n→+∞.

Proof. Based on Equation VIII.2, without secondary market, the optimal order for player iis x^∗_i =F_Q⁻¹

1− ^c

c⁰_i

. It can be also denoted as x^∗_i =EQ(Q)±Θ(1). So the aggregated

order isOD_2nd =nEQ(Q)±Θ(n), and

OD_2nd−OD_#

=Θ(n)asn→+∞.

Theorem VIII.2.7. The difference between the aggregated orderswithsecondary market and the optimal aggregated orders is upper bounded by O(¹_cn·ψ(n)), i.e.

SW_#−SW_2nd = O(¹_cn·ψ(n))as n→+∞, whereψ(n) = ^{max(xmax,qmax)}

min(σ√ n,π(n)).

Proof. Based on Equation VIII.40 in the proof of Theorem VIII.2.5, we could know that SW_2nd=−cnEQ(Q)±O(n·ψ(n)). As the price in the primary market iscand the expected price in a secondary market is also c. So the overall quantity of resources in the two markets isOD_2nd = ^|SW_c^2nd| =nEQ(Q)±O(¹_cn·ψ(n)). So we could get|OD_2nd−OD_#|= O(¹_cn·ψ(n))asn→+∞.