“Very Nice” Trivial Equilibria in Strategic Market Games ∗ Francesca Busetto† Giulio Codognato‡ March 2004

Abstract Following Shapley (1976), we study the problem of the existence of a Nash Equilibrium (NE) in which each trading post is either active or “legitimately” inactive, and we call it a Shapley NE. We consider an example of an exchange economy, borrowed from Cordella and Gabszewicz (1998), which satisfies the assumptions of Dubey and Shubik (1978), and we show that the trivial equilibrium, the unique NE of the associated strategic market game, is not “very nice,” in the sense that it is not “legitimately” trivial. This result has the more general implication that, under the Dubey and Shubik’s assumptions, a Shapley NE may fail to exist. Journal of Economic Literature Classification Numbers: C72, D51.

1

Introduction

In this paper, we consider the problem of the existence of a Nash Equilibrium (NE) within the research program for strategic market games founded by Shubik (1973), Shapley (1976), and Shapley and Shubik (1977). ∗

We would like to thank Enrico Minelli for comments and suggestions. Dipartimento di Scienze Economiche, Universit`a degli Studi di Udine, Via Tomadini 30, 33100 Udine, Italy. ‡ Dipartimento di Scienze Economiche, Universit`a degli Studi di Udine, Via Tomadini 30, 33100 Udine, Italy. †

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In particular, we focus on a prototypical strategic market game, originally proposed by Shubik (1973) and Shapley (1976), in which traders can offer each commodity for money in a trading post, and prices are determined as the ratio between total offers and total money bids. These authors observed that a trivial NE - that is, a NE in which every trading post is inactive, in the sense that nothing is offered on either side of it - always exists. Shapley (1976) also considered the opposite situation where, at a NE, no trading post is allowed to be inactive. This author observed that this equilibrium configuration is too demanding, as it rules out the possibility that some trading post is “legitimately” inactive, that is, even if it were “open for business,” it would attract no offer and money bid. He formalized this idea by introducing the crucial concept of a virtual price and stressed (Shapley (1976), p. 171): “Our goal is a theorem that asserts [...] that a NE exists in which each commodity has either an actual price [...], if actively traded, or a virtual price [...], if not.” (See also Shapley and Shubik (1977), p. 365). Here, we shall call such an equilibrium a Shapley NE. Dubey and Shubik (1978) proposed to show the existence of a Shapley NE for the strategic market game introduced in Shubik (1973) and Shapley (1976) by using an auxiliary equilibrium notion called Equilibrium Point (EP). An EP is a pair consisting of a NE of the strategic market game and a price system, which are the limits of sequences of Nash Equilibria and prices generated by perturbing the game itself. Dubey and Shubik (1978) called “nice” the NE corresponding to an EP. They proved the existence of an EP under rather general assumptions on traders’ endowments and preferences. Moreover, in their Remark 2, they characterized an EP in such a way that it turns out to be a Shapley NE. Therefore, Dubey and Shubik’s existence theorem for an EP, together with Remark 2, implies the existence of a Shapley NE. Cordella and Gabszewicz (1998) pointed out a class of exchange economies, satisfying the Dubey and Shubik’s assumptions, where the unique NE of the associated strategic market game is the trivial equilibrium. They qualified this equilibium as “nice,” in that they proved that it is the unique limit of sequences of Nash Equilibria generated by perturbing the game. In this paper, we consider a particular exchange economy within the class studied by Cordella and Gabszewicz (1998) and we show that the pair consisting of the trivial equilibrium and the corresponding limit prices is the only EP of the associated strategic market game. We show that this unique 2

EP is not a Shapley NE, thereby contradicting the claim that an EP is always a Shapley NE. This raises the question whether the unique “nice” trivial equilibrium is also “very nice,” in the sense that it is “legitimately” trivial. Our answer is no. Indeed, we show that there is no virtual price corresponding to the trivial equilibrium of the strategic market game associated with our exchange economy. This result has the more general implication that, under the Dubey and Shubik’s assumptions, a Shapley NE may fail to exist. The paper is organized as follows. In Section 2, we introduce the strategic market game. In Section 3, we show that an EP may not be a Shapley NE and that a Shapley NE may fail to exist. In Section 4, we conclude by outlining some perspectives for further research.

2

The model

In this section, we introduce the version proposed by Dubey and Shubik (1978) of a strategic market game originally analyzed by Shubik (1973) and Shapley (1976). Consider an exchange economy with n traders and m + 1 commodities, where the (m + 1)th commodity plays the role of money. The initial bundle m+1 of trader i is a vector ai ∈ R+ , where aij is the amount of commodity P i j available to i. We assume that m+1 i=1 a À 0. A trader i is said to be “j-furnished” if aij > 0, j = 1, . . . , m, and “moneyed” if aim+1 > 0. m+1 The preferences of i are described by an utility function, ui : R+ → R, satisfying the following assumption. m+1 Assumption 1. ui : R+ → R is continuous, nondecreasing, and concave.

We shall say that trader i “desires” commodity j if ui (xi ) is an increasing function of the variable xij , for any fixed choice of the other variables xik , k 6= j. Consider now the following strategic market game, Γ, associated with this exchange economy. For each commodity j = 1, . . . , m, there is a trading post, Mj , where commodity j is exchanged for money. The strategy set of trader i is m i m i , qj ≤ aij , ¯bi ≤ aim+1 }, , b ∈ R+ S i = {(q i , bi ) : q i ∈ R+

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where qji is the quantity of commodity j that i offers for sale in the trading post Mj , bij is i’s money bid in the same trading post, and ¯bi stands for Pm i j=1 bj . Trading posts can be characterized as follows. Definition 1. A trading post Mj is said to be active if q¯j > 0 and/or ¯bj > 0, and inactive if q¯j = ¯bj = 0. Let S = S 1 × · · · × S n and S −i = S 1 × · · · × S i−1 × S i+1 × · · · × S n . Moreover, let s, si , and s−i be elements of S, S i , and S −i , respectively. For m each s ∈ S, p(s) ∈ R+ is a price vector determined according to the following rule:   ¯bj , if q¯ > 0, j q¯j pj (s) = (1)  0 if q¯j = 0, P P for each j = 1, . . . , m, where q¯j = ni=1 qji and ¯bj = ni=1 bij . Then, for each m+1 s ∈ S, the final bundle of trader i is a vector xi (s) ∈ R+ such that

xij (s)

 

aij − qji + =  ai − q i j j

bij , pj

if pj > 0, if pj = 0,

(2)

for j = 1, . . . , m, and xim+1 (s)

=

aim+1

−

m X

bij

+

j=1

m X

qji pj .

(3)

j=1

Finally, the payoff of trader i is π i (s) = ui (xi (s)).

(4)

We define now the notion of a Nash Equilibrium of Γ. Definition 2. An sˆ ∈ S is a Nash Equilibrium (NE) of Γ if, for each i = 1, . . . , n, π i (ˆ si , sˆ−i ) ≥ π i (si , sˆ−i ), for all si ∈ S i . The problem of the existence of a NE is at the heart of the research program for strategic market games founded by Shubik (1973), Shapley (1976), and Shapley and Shubik (1977). With reference to the model described above, Shapley (1976) noticed that an s such that si = (0, 0), i = 1, . . . , n, 4

is always a NE. For this reason, in the literature, it has been called a trivial equilibrium. On the opposite side, this author considered the existence of a “full-fledged” NE, in which all trading posts Mj are active. He observed, however, that this characterization of a NE is too demanding, since it rules out the possibility that some trading post may be “legitimately” inactive in the sense that, even if it were “opened for business,” it would attract no trade. Shapley (1976) formalized this idea by defining the concept of a virtual price, which we shall now introduce. Given an s ∈ S and the real numbers vj ≥ 0, δj > 0, let π i (s, δj , δj vj ) be the trader i’s payoff corresponding to s, when some outside agency offers the quantity δj for sale in the trading post Mj , makes the money bid δj vj in the same trading post, and prices and final bundles are determined according to (1), (2), and (3). Then, we have the following definition. Definition 3. Let sˆ ∈ S be a NE in which Mj is inactive. A real number vj ≥ 01 is said to be a virtual price for Mj if, for each i = 1, . . . , n, π i (ˆ s, δj , δj vj ) ≥ π i (si , sˆ−i , δj , δj vj ), for all si ∈ S i and for all δj > 0. Using the notion of a virtual price, Shapley (1976) proposed the following equilibrium concept for Γ, which we shall call a Shapley NE. l Definition 4. A pair (ˆ s, pˆ), where sˆ ∈ S and pˆ ∈ R+ , is a Shapley NE of Γ if sˆ is a NE of Γ and, for each j = 1, . . . , l, pˆj = pj (ˆ s), if Mj is active, pˆj = vj , if Mj is inactive, where vj is a virtual price for Mj .

One of the main goals of the Shapley and Shubik’s research program for strategic market games consists in showing the existence of a Shapley NE. 1

It is worth noticing that Definition 3 allows virtual prices to be equal to zero. This possibility is not explicitly ruled out in the model proposed by Shapley (1976), where “actual” prices are not defined for inactive trading posts. On the contrary, in the Dubey and Shubik’s version of the model, which we use here, “actual” prices for inactive trading posts coincide with virtual prices when the latter are equal to zero.

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3

The nonexistence of a Shapley Nash Equilibrium

In order to show the existence of a Shapley NE, Dubey and Shubik (1978) introduced the following further assumption. Assumption 2. (i) All traders desire money; (ii) for any commodity j 6= m + 1, there are at least two moneyed traders who desire j, and at least two j-furnished traders. Their existence proof is based on an auxiliary equilibrium notion, called Equilibrium Point, which we shall describe now. In so doing, we shall follow not only the original work by Dubey and Shubik (1978), but also Amir et al. (1990), where the concept of an Equilibrium Point is more explicitly defined in terms of a pair of strategies and prices. Denote by ² Γ a modification of the game Γ where some outside agency places a fixed amount ² > 0 on both sides of each trading post Mj . Accordingly, given an s ∈ S, let ² pj (s) be a price vector determined as in (1) and ² i x (s), ² π i (s) i’s final bundle and payoff determined as in (2), (3), and (4). Then, we have the following definition. m Definition 5. The pair (˜ s, p˜), where s˜ is a NE of Γ and p ∈ R++ , is an ∞ ²l ∞ Equilibrium Point (EP) of Γ if there exist sequences {²l }l=1 , { s˜}l=1 , and ²l {²l p˜}∞ ˜ is a NE of ²l Γ and ²l s˜ → s˜, l=1 such that (i) ²l > 0 and ²l → 0, (ii) s (iii) ²l p˜ = p(²l s˜) and ²l p˜ → p˜.

Dubey and Shubik (1978) called “nice” the NE s˜ corresponding to an EP (˜ s, p˜) of Γ. With reference to the EP notion, they were able to show the following theorem. Theorem. Under Assumptions 1 and 2, an EP of Γ exists. Moreover, in their Remark 2, they asserted (Dubey and Shubik (1978), p. 13): “If at the EP [...] the aggregate bids and offers in any trading post are zero, we still associate the virtual price [...] with that trading post.” This characterization of an EP implies that it is a Shapley NE. Using our notation, Dubey and Shubik’s Remark 2 can thus be expressed by means of the following claim.

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Claim. Under Assumptions 1 and 2, if a pair (˜ s, p˜) is an EP of Γ, then it is a Shapley NE of Γ. The Theorem and the Claim together imply the existence of a Shapley NE of Γ. Following Dubey and Shubik (1978), Cordella and Gabszewicz (1998) pointed out a class of exchange economies satisfying Assumptions 1 and 2, where the unique NE of the game Γ is the trivial equilibrium and they showed that it is the limit of a sequence of Nash Equilibria of the perturbed games ² Γ. In this sense, the trivial equilibrium can be qualified as “nice,” even though they did not develop a complete analysis in terms of an EP, since they neglected considering prices. Here, we complete the analysis of Cordella and Gabszewicz (1998) by considering a specific exchange economy within their class and showing that there is a unique EP for the associated game Γ. Moreover, we show that this unique EP is not a Shapley NE. Example. Consider an exchange economy satisfying Assumptions 1 and 2, where m = 1, T = {1, 2, 3, 4}, ai = (1, 0), ui (x) = 23 x1 + x2 , for i = 1, 2, ai = (0, 1), ui (x) = x1 + 32 x2 , for i = 3, 4, and the associated game Γ. Then, the pair (˜ s, p˜1 ) where s˜i = (0, 0), i = 1, . . . , 4, and p˜1 = 1, is the unique EP of Γ and it is not a Shapley NE. Proof. It is straightforward to verify that, for each ² > 0, the ² s˜ such that ² i s˜ = (², 0), i = 1, 2, ² s˜i = (0, ²), i = 3, 4, is the unique NE of ² Γ which lies in the interior of S. Then, for any sequence {²l }∞ l=1 such that ²l → 0, we have ²l i that the sequence {²l s˜}∞ , with s ˜ = (² , 0), i = 1, 2, ²l s˜il = (0, ²l ), i = 3, 4, l l=1 l satisfies s˜l → s˜, with s˜i = (0, 0), i = 1, . . . , 4, and the sequence {²l p˜1 }∞ l=1 , 3²l ²l ²l with p˜1 = 3²l , satisfies p˜1 → p˜1 , with p˜1 = 1. Moreover, it is possible to show - but we omit the details - that the limit of all the other convergent sequences determined according to Definition 5 is the pair (˜ s, p˜1 ). Then, this pair turns out to be the unique EP of Γ. In order to prove that this EP is not a Shapley NE, we have to show that p˜1 = 1 is not a virtual price. Given an s ∈ S and a real number δ1 > 0, the traders’ payoffs, when an outside agency offers for sale the quantity δ1 and makes the money bid δ1 p˜1 = δ1 in the trading post M1 , are b3 + b4 + δ1 2 , i = 1, 2, π i (s, δ1 , δ1 ) = (1 − q i ) + qi 1 3 q + q 2 + δ1 7

q 1 + q 2 + δ1 2 + (1 − bi ), i = 3, 4. b3 + b4 + δ1 3 Then, it is immediate to see that π i (s, δ1 , δ1 ) = bi

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∂π i (˜ s, δ1 , δ1 ) > 0, i = 1, 2, ∂q i ∂π i (˜ s, δ1 , δ1 ) > 0, i = 3, 4, ∂bi for all δ1 > 0. This implies that p˜1 = 1 is not a virtual price. Thus, the pair (˜ s, p˜1 ) is not a Shapley NE. The example above contradicts the Claim and shows that there may exist a “nice” NE in which an inactive trading post is not “legitimately” inactive. This leads us to call “very nice” a NE sˆ corresponding to a Shapley NE (ˆ s, pˆ) of Γ, that is a NE in which all inactive trading posts are “legitimately” inactive. The question arises whether the unique NE of the Example, the trivial equilibrium, is also “very nice.” This, in turn, poses the more general question whether, under Assumptions 1 and 2, a Shapley NE of Γ always exists. The following proposition provides a negative answer. Proposition. There is an exchange economy satisfying Assumptions 1 and 2 for which there exists no Shapley NE (ˆ s, pˆ) of the associated game Γ. Proof. Consider the same exchange economy as in the example above and the associated game Γ. Given an s ∈ S and the real numbers v1 ≥ 0, δ1 > 0, the traders’ payoffs, when, in the trading post M1 , an outside agency offers for sale the quantity δ1 and makes the money bid δ1 v1 , are 2 b3 + b4 + δ1 v1 π i (s, δ1 , δ1 v1 ) = (1 − q i ) + qi 1 , i = 1, 2, 3 q + q 2 + δ1 π i (s, δ1 , δ1 v1 ) = bi

q 1 + q 2 + δ1 2 + (1 − bi ), i = 3, 4. 3 4 b + b + δ1 v1 3

As shown by Cordella and Gabszewicz (1998), there is a unique NE of Γ, which we denote by sˆ, and it is given by sˆi = (0, 0), i = 1, . . . , 4. Then, it is immediate to see that

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∂π i (ˆ s, δ1 , δ1 v1 ) 2 = v1 − > 0, i = 1, 2, i ∂q 3 for all v1 >

2 3

and for all δ1 > 0, ∂π i (ˆ s, δ1 , δ1 v1 ) 1 2 = − > 0, i = 3, 4, i ∂b v1 3

for all 0 < v1 < 23 and for all δ1 > 0. This implies that no real number v1 > 0 can be a virtual price. It remains to consider the case where v1 = 0. For each δ1 > 0, we have π i (ˆ s, δ1 , 0) =

2 2 < δ1 + (1 − bi ) = π i (si , sˆi−1 , δ1 , 0), 3 3

for all 0 < bi < δ1 , i = 3, 4. Therefore, there exists no Shapley NE for the game Γ associated with the exchange economy of the Example, and this completes the proof.

4

Conclusions

In this paper, we have considered the problem of the existence of a NE for strategic market games, in the vein of Shapley (1976). In particular, this author raised the issue of proving the existence of a NE with “legitimately” inactive trading posts, which we have called a Shapley NE. We have analyzed the approach to this issue proposed by Dubey and Shubik (1978) with reference to the prototypical strategic market game introduced in Shubik (1973) and Shapley (1976). This approach is based on the proof of the existence of an EP, an auxiliary equilibrium notion, which they characterized as a Shapley NE. Using a framework borrowed from Cordella and Gabszewicz (1998), we have shown, through an example, that an EP may not be a Shapley NE. Furthermore, we have obtained the more general result that, under the assumptions of Dubey and Shubik (1978), a Shapley NE may fail to exist. This has made clear that, in order to prove the existence of a Shapley NE, more restrictive assumptions on traders’ endowments and preferences than those made by Dubey and Shubik (1978) are to be introduced. A way 10

to tackle this problem could be to complete the approach proposed by these authors, by explicitly showing that an EP is a Shapley NE. Another possible route to follow is abandoning the very notion of a Shapley NE and showing the existence of a NE in which no market can be, even if “legitimately,” inactive. This approach was applied by Peck et al. (1992) to a different prototypical strategic market game. This way they proved the existence of interior Nash Equilibria - that is Nash Equilibria in which all offers and bids are positive - under the assumption that traders’ endowments and indifference curves are in the interior of the commodity space.

References [1] Amir R., Sahi S., Shubik M., Yao S. (1990), “A strategic market game with complete markets,” Journal of Economic Theory 51, 126-143. [2] Cordella T., Gabszewicz J.J. (1998), ““Nice” trivial equilibria in strategic market games,” Games and Economic Behavior 22, 162-169. [3] Dubey P., Shubik M. (1978), “The noncooperative equilibria of a closed trading economy with market supply and bidding strategies,” Journal of Economic Theory 17, 1-20. [4] Peck J., Shell K., Spear S.E. (1992), “The market game: existence and structure of equilibrium,” Journal of Mathematical Economics 21, 271299. [5] Shapley L.S. (1976), “Noncooperative general exchange,” in S.A.Y. Lin (ed), Theory of measurement of economic externalities, Academic Press, New York. [6] Shapley L.S., Shubik M. (1977), “Trade using one commodity as a means of payment,” Journal of Political Economy 85, 937-968. [7] Shubik M. (1973), “Commodity money, oligopoly, credit and bankruptcy in a general equilibrium model,” Western Economic Journal 11, 24-38.

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