Introduction

Games

2073-4336

Molecular Diversity Preservation International (MDPI)

10.3390/g2020209

games-02-00209

Article

Competing in Several Areas Simultaneously: The Case of Strategic Asset Markets

Nermuth

Manfred

Department of Economics, University of Vienna, Hohenstaufengasse 9, A-1010 Vienna, Austria; E-Mail: manfred.nermuth@univie.ac.at; Tel.: +43 1 427737440; Fax: +43 1 4277 9374

2011

12 04 2011

2 2 209 234 20 12 2010 16 03 2011 07 04 2011

2011

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/.)

We characterize the structure of Nash equilibria for a certain class of asset market games. In equilibrium, different assets have different returns, and (risk neutral) investors with different wealth hold portfolios with different structures. In equilibrium, an asset's return is inversely related to the elasticity of its supply. The larger an investor, the more diversified is his portfolio. Smaller investors do not hold all the assets, but achieve higher percentage returns. More generally, our results can be applied also to other “multi-market games” in which several players compete in several arenas simultaneously, like multi-market Cournot oligopolies, or multiple rent-seeking games.

asset markets Nash Equilibrium multigames

1. Introduction 1.1. Motivation

There are many situations in which a number of players play several different games (interact in different markets) simultaneously, and what a player can do in one game is constrained by what he does in the others (so that the games cannot be analyzed separately). Typically, there is a resource constraint: Each player has a limited amount of some resource (time, money, effort, capacity,…) and must decide how much of it to commit to the various “markets” in order to maximize his overall payoff.

Examples include: Strategic investors (with limited budget) in certain asset markets (the subject of this paper), multimarket Cournot oligopoly (with limited capacity), but also political contests where several candidates (with limited time) compete in several constituencies, multiple rent-seeking, etc.

We study such a situation of “multimarket interaction” for the special case of asset market games of the type considered in [1–3]1. These works in turn build on older work on market games of the kind introduced in [6], and especially on work on strategic financial markets, like [7].

The present study was directly motivated by the model in [1]: Here investors decide how to allocate their funds over a number of different assets, which are in constant (unit) supply. Prices are set so as to equate demand and supply. [1] noted that the unique Nash equilibrium of the game coincides with the unique evolutionarily stable strategy (ESS), and also coincides with the “competitive outcome”. While a relationship between evolutionary stability and competitive outcomes has also been observed in other contexts ([8,9]), and thus is not too surprising, the coincidence of competitive outcomes and Nash equilibria is rather striking. It seems that the familiar tension (e.g., in Cournot oligopoly) between price-taking and strategic behavior is absent from these asset markets2. It was this observation which originally motivated the present study.

It turns out that a slight modification of the model suffices to remove this anomaly: Rather than constant (inelastic) asset supply, we allow elastic supply, such that different assets may have different elasticities. This recognizes the fact that in an asset market usually only a few investors (e.g., very large, professionally managed funds) will act strategically, that is, be aware that their transactions may influence prices, and take this influence into account for their investment decisions. Only these large, strategic investors will act as players in our market game. Even if the total supply of an asset is really constant, part of it will usually be held by other, nonstrategic market participants, and these may be more or less willing to sell, depending on the price. We model these traders in the simplest possible way by assuming that their aggregate behavior is described by an upward-sloping (possibly constant) supply function (one for each asset). Given this, Nash and competitive outcomes no longer coincide (see Section 1.2).

We also assume that the markets for the various assets are sufficiently separated, so that the supply of each asset depends only on its own price. While such an assumption is certainly restrictive3, this permits us to obtain very detailed information about the structure of equilibrium, which would probably be impossible otherwise.

We note also that uncertainty plays no formal role in our analysis. Following part of the literature which motivated this study, we assume that the players are risk neutral, so that only expected payoffs matter. Thus the results in Sections 1–4 of [1] remain true even with elastic supply, as long as all elasticities are the same (Theorem 4.2).

Remark

Even though our analysis was originally motivated by a financial markets literature, such markets are probably not the best examples for our theory. Indeed, in financial markets an important role is played by factors which are not modeled in the present paper, like differential information, risk aversion, short selling, interdependence of asset prices, etc. Combining such features with strategic behavior is quite difficult and requires a different sort of analysis (cf. [5,7]).

Instead of financial asset markets, it might be better to think of real assets. For example, the players could be a group of large investment funds buying real estate in various cities (“markets”), like New York, London, Hongkong, … Each player has a certain budget and must decide how much to invest in the various cities; moreover the players realize that their investments may influence the prices. Alternatively4 the players might be large strategic traders who compete in buying various input factors and sell them on the world market, at a given world market price.

1.2. Summary of Results

Given the exogenous supply functions (one for each asset) the strategic players decide how much of their available funds to invest in the various assets. Prices are then set to equate supply and demand in each market, exactly as in the classical market games with fixed supply. But note that the game is no longer constant-sum.

The main contribution of the present paper is a detailed characterization of the structure of the possible Nash equilibria of this asset market game, for the non-symmetric case. The results can be summarized as follows: In the symmetric case, when all supply functions have the same elasticity, then there exists a unique Nash equilibrium. It is symmetric and coincides with both the “competitive” and the ESS outcomes (here “competitive” is of course to be understood as a situation in which every investor maximizes profits, taking prices–not supply functions—as given). In this equilibrium, prices are proportional to expected payoffs (prices are “fair”, or “correspond to fundamentals”) and all investors achieve the same rate of return on their capital. This includes the constant supply situation considered in the literature quoted above. Intuitively, when the supply of all assets is constant, the market game is constant-sum, and a player can increase his payoff only at the expense of the others; therefore the maximization of absolute payoffs (Nash) coincides with the maximization of relative payoffs (ESS)5. We show that this continues to hold for variable supply, as long as the supply conditions for all assets are essentially the same (Theorem 4.2).

But the point of the paper is the non-symmetric case where the supply conditions for different assets are genuinely different. Then Nash equilibria are not symmetric, and neither ESS nor competitive. Prices are not fair, and different assets have different return rates. More precisely, the lower the elasticity of supply of an asset, the higher its return. Larger investors are more diversified, with the largest investor holding positive quantities of all assets, but smaller investors buy only some of the assets (the smaller, the fewer). Moreover, larger investors necessarily hold relatively more low-yielding assets and achieve lower average rates of return at equilibrium than small investors (Theorem 4.1); this may be termed the “curse of size”.

These somewhat counterintuitive deviations from the competitive outcome, i.e. different rates of return across assets and/or investors, have nothing to do with market imperfections or other reasons like different degrees of risk aversion etc., but come from the heterogeneity of supply, combined with the strategic interplay among large and small investors: at equilibrium, every investor equalizes the marginal, not the average, rates of return of all assets which he holds in positive quantity Since marginal rates differ from average ones, and also across investors due to their different wealth, we obtain these heterogeneous portfolios.

Equilibrium is unique in the “symmetric” cases (symmetry w.r.t. assets and/or investors, Theorem 4.2 and Theorem 4.3); in general, we can only prove that there exists at most one Nash equilibrium at which all investors hold all assets (Proposition 4.2; there may exist no such equilibrium).

We also consider competitive and ESS outcomes separately. There always exists a unique ESS, and a unique competitive rate of return (the same for all assets). At the ESS, all investors achieve exactly the competitive rate6. At any Nash equilibrium that is not ESS, they achieve strictly more (Theorem 5.1, Theorem 4.4).

The paper is organized as follows. Section 2 introduces the basic model, in Section 3 we study competitive allocations and prove the existence of Nash equilibrium, Section 4 contains the main results, and in Section 5 we study evolutionarily stable strategies. Most proofs, except very short ones, are in the appendix.

2. Notation and Definitions

We consider an asset market of the kind studied in [1–3]. There are i = 1, 2, …, N risk-neutral investors (N ≥ 2), and k = 1, 2,…,K assets (K ≥ 2). The initial endowment (with money) of investor i is Wⁱ > 0, and the total money endowment is W: = Σ_iWⁱ. The (expected) monetary payoff per unit of asset k is E_k> 0, and S_k(p_k) is the supply function for asset k, where p_k ≥ 0 denotes the price (per unit) of asset k. We allow arbitrary supply functions, subject only to the condition that the price elasticity of supply be non-increasing. We denote the supply elasticity of asset k by η k = H k ( p k ) = p k S k ′ ( p k ) S k ( p k ) for p k > 0 and assume, for ∀k:

S.1. The supply function S_k(p_k) is continuous and nondecreasing for p_k ≥ 0, and strictly positive for p_k > 0.

S.2. The supply function S_k(p_k) is twice continuously differentiable (possibly with infinite slope at p_k = 0)7.

S.3. The elasticity H_k(p_k) is non-increasing on (0, ∞).

Remark

It may be that S_k(0) = 0 or S_k(0) > 0; we will show (cf. Lemma A.2 in the appendix) that the latter case occurs if and only if supply is constant, S_k(p_k) = S̄_k> 0 ∀p_k ≥ 0. This is the situation considered in the literature quoted above8. To justify a variable supply in the strategic N-player game among “big” investors which we are going to study, we may imagine that the asset is also held by a large number of small, nonstrategic market participants. These traders are simply willing to sell more of an asset if its price goes up; their aggregate behavior is captured by the supply functions S_k.

Each investor i invests his whole wealth Wⁱ in the K available assets, i.e. he chooses a vector w i = ( w 1 i , … , w K i ) in his budget set (1) B i = B i ( W i ) = { w i ∈ ℝ + K | ∑ k w k i = W i }where w k i ≥ 0is the amount of money invested in asset k by investor i.

The set Bⁱ is i's strategy space; it is a nonempty, compact and convex subset of ℝ + K. The joint strategy space is B : = ∏ i B i ⊂ ℝ + K N. A strategy profile is denoted by w = (w¹,…w^N) ∈ B. Given w, we write w k : = ∑ i w k i for the total amount of money invested in asset k by all investors (wⁱ is a vector, but w_k is a scalar!). If w_k> 0 we say that market k is active (at w).

Alternatively (and equivalently), the behavior of an investor i can be described in percentage terms, i.e. by α i = ( α 1 i , … α K i ), where α k i = w k i / W i denotes the fraction of investor i's wealth invested in asset k. This formulation makes the game appear more symmetric and is useful in certain contexts (e.g., to study evolutionary stability, see Section 5). The vector αⁱ describes how investor i allocates his money among the various assets (30% in German real estate, 3% in Gold, etc.); for convenience we refer to αⁱ also as investor i's portfolio. Similarly, we use the word “market portfolio” for the vector (w₁/W,…w_N/W), describing the percentage allocation of total wealth W over the K asset types (in the finance literature, the term “market portfolio” has a slightly different meaning, but this should cause no confusion).

For p_k > 0, we denote by r_k: = E_k/p_k the (gross) rate of return per dollar of asset k. In a competitive equilibrium, with price-taking risk-neutral investors, r_k must be the same for all assets (cf. Section 3.1).

Remark

even though, in the game to be considered below, investors are constrained to choose wⁱ ∈ Bⁱ, many of the following considerations do not depend on this restriction, but are valid for arbitrary nonnegative wⁱ ∈ ℝ^K. We will therefore, whenever appropriate, pay no attention to the budget constraints and consider arbitrary strategy profiles w ∈ ℝ + K N.

Given w ∈ ℝ + K N, the prices p_k are determined so as to clear markets, i.e. such that ∀k

(2) p k S k ( p k ) = w k

It is easy to see (cf. Lemmas A.1, A.3) that this defines a unique price p_k = P_k(w_k) for every w_k ≥ 0; and that the price function p_k = P_k(w_k) is differentiable and strictly increasing, with P_k(0) = 0, lim_{w_k→∞} P_k(w_k) = ∞. Therefore the rate of return, (3) r k = R k ( w k ) : = E k P k ( w k ) = E k S k ( P k ( w k ) ) w k is also a differentiable function of w_k, and strictly decreasing in w_k, with lim_{w_k→}₀ R_k(w_k) = ∞ and lim_{w_k→∞} R_k(w_k) = 0. Note in particular that if the total investment in asset k goes to zero, w_k → 0, then the expected rate of return R_k = R_k(w_k) becomes arbitrarily large. This will ensure that in Nash equilibrium all markets are active.

We conclude this section with a formula that will be useful in the sequel. Note that by definition (2), P_k(w_k)S_k(P_k(w_k)) ≡ w_k. Totally differentiating gives: P k ′ ( w k ) S k ( P k ( w k ) ) + P k ( w k ) S k ′ ( P k ( w k ) ) P k ′ ( w k ) ≡ 1 ⇔ P k ′ S k [ 1 + η k ] ≡ 1

Using S_k = w_k/P_k we obtain: (4) 0 < 1 1 + H k [ P k ( w k ) ] ≡ P k ′ ( w k ) w k P k ( w k ) ≤ 1

Expression (4) is the elasticity of the price p_k = P_k(w_k) with respect to the total amount of money w_k invested in asset k. It lies between zero and one because η_k ≥ 0 by S.1 and is equal to one if and only if the supply elasticity η_k = H_k[P_k(w_k)] is zero (i.e., supply is constant).

3. The Market Game

Given a strategy profile w ∈ ℝ + K N, the amount of asset k allocated to investor i is given by: (5) x k i ( w ) : = { w k i / P k ( w k ) if w k > 0 ( ⇒ p k > 0 ) 0 if w k = 0 ( market k is not active )

If market k is active we may also write (6) x k i ( w ) = w k i w k S k ( P k ( w k ) ) ( w k > 0 )

If i is the only investor who holds asset k ( w k i = w k ), then (7) x k i ( w ) = { S k ( P k ( w k ) ) if w k i > 0 or S k ( 0 ) = 0 0 [ ≠ S k ( 0 ) ] if w k i = 0 and S k ( 0 ) ≠ 0 with a discontinuity at w k i = 0 if S_k(0) > 0. The payoff of investor i is then given by (8) π i ( w ) = ∑ k E k x k i ( w ) = ∑ w k > 0 k w k i R k ( w k )

These data define the asset market game G among the N investors, with strategy spaces Bⁱ and payoff functions πⁱ.

In this formulation the strategies and payoffs of large and small investors are not directly comparable, but we can make them so by expressing everything in percentage terms, i.e. by dividing both the invested amounts w k i and the payoff πⁱ of an investor i by his initial wealth Wⁱ. This gives a strategically equivalent game Ḡ as follows. The strategy of investor i is his portfolio α i = ( α 1 i , … , α K i ) = ( W ) − 1 ⋅ ( w 1 i , … , w K i ) his strategy space is the (K − 1)-dimensional unit simplex ∆^K, a joint strategy is written α = (α¹, …, α^N), and the payoff of investor i in the modified game Ḡ is his return per dollar (9) r i = π ¯ i ( α ) = ∑ α k i > 0 k α k i R ¯ k ( α ) = π i ( w ) W i where R ¯ k ( α ) : = R k ( ∑ i W i α k i ); and ∑ i W i α k i = ∑ i w k i = w k is the total amount invested in asset k if the investors use the joint strategy α.

This formulation of the game is quite natural. Indeed, investors frequently describe their strategy by stating how many percent of their total funds they invest in various types of assets, and they describe their payoffs by stating the percentage return on their portfolio (e.g., 4.5% p.a.), not the absolute quantities. We will use the two equivalent formulations G and Ḡ interchangeably, according to convenience, and refer to both G and Ḡ as the “asset market game”.

Our main interest will be in determining the structure of the Nash equilibria of this game, corresponding to strategic (fully rational) behavior of the agents (Section 4). But we will also consider other solution concepts: Competitive outcomes corresponding to price-taking behavior, and evolutionarily stable strategies (ESS) in the sense of [11], which are motivated by certain types of boundedly rational behavior (imitation) (Section 5). The relationships between these solution concepts will also be clarified. Moreover, it is easy to see that the game G is constant-sum if and only if all supply functions S_k are constant (see Lemma A.4 in the appendix).

Given a strategy profile w ∈ B, the expected payoff from asset k, E_kS_k = E_kS_k[P_k(w_k)], and the total expected payoff, E(w) : = Σ_k E_kS_k, are determined. We say that investor i follows the proportional investment rule if the amount invested by him in each asset k is proportional to the expected payoff E_kS_k of this asset, i.e. if there is γⁱ > 0 such that w k i = γ i E k S k ∀ k (or α k i = E k S k / E ( w ); “investing according to the fundamentals”).

3.1. Competitive Allocations

A profile w = (w¹,…w^N) (resp. the corresponding asset allocation) is called competitive if all assets have the same rate of return, i.e. if there exists r̂ such that for all k (10) R k ( w k ) = r ^

In this case p k = 1 r ^ E k ∀ k, i.e. prices are proportional to expected payoffs. We also say that prices are fair. Clearly, in a competitive allocation, the common rate of return r̂ is equal to the total payoff divided by the total initial money endowment (11) r ^ = E ( w ) W (to see this, use (3) to write w_k= r̂⁻¹E_kS_k and sum over k).

Lemma 3.1.

There exists a unique competitive rate of return r̂ > 0. It depends only on the total money endowment W, but not on the distribution of wealth W¹, … W^N.

Proof

For r > 0, define w_k(r) > 0 by R_k[w_k(r)] = r. The properties of the return function R_k(·) imply that w_k(r) is well defined and strictly decreasing in r (from ∞ to 0); hence there exists a unique r̂ > 0 such that Σ_kw_k(r̂)) = W.

Define ŵ_k by R_k(ŵ_k) = r̂. In competitive equilibrium the return rate r̂, the prices p̂_k = E_k/r̂, the profits π i ( w ) = ∑ k w k i R k = W i r ^, and the amounts ŵ_k are uniquely determined, but not the asset allocation. Indeed, from the viewpoint of a price-taking investor i, any strategy w i = ( w 1 i , … w K i ) ∈ B i is profit maximizing, since all assets yield the same return. Thus there are infinitely many competitive allocations, characterized by the condition that the total amounts w k = ∑ i N w k i invested in the various assets satisfy w_k = ŵ_k. Among these, a special role is played by the proportional investment rule.

Lemma 3.2.

(i) There exists a unique profile ŵ ∈ B in which all investors follow the proportional investment rule. This profile is competitive and is given by w ^ k i = w ^ k W i W, where ŵ_k is given by R_k(ŵ_k) = r̂. (ii) Let w ∈ B be a competitive profile such that all investors hold the same portfolio αⁱ = α^j ∀i,j. Then w = ŵ and αⁱ = α̂ ∀i, where α ^ = ( α ^ 1 , … α ^ K ) : = 1 W ( w ^ 1 , … w ^ K ) ∈ Δ K is the market portfolio corresponding to the profile ŵ ∈ ℝ.

Let us call a profile w = (w¹, …w^N) ∈ B symmetric9 if all investors hold the same portfolio, i.e. if αⁱ = α^j for all i, j, where αⁱ = (1/Wⁱ)wⁱ. Lemma 3.2 (ii) says that ŵ is the only symmetric competitive profile. We shall see below that this “proportional competitive profile” ŵ has certain special properties. In particular, if a Nash equilibrium allocation w is competitive (this is not the case in general), then it coincides with ŵ (Propostion 4.1). Moreover, ŵ is the unique profile that is evolutionarily stable in the sense of [11] (Theorem 5.1).

3.2. Nash Equilibrium

A profile w* = (w*1,…w*^N) ∈ B is a Nash equilibrium of the game G if for all investors i = 1,…N π i ( w ∗ ) ≥ π i ( w i , w ∗ − i ) ∀ w i ∈ B i where (wⁱ, w^*−i) denotes the profile w^* with i's strategy w^*i replaced by wⁱ.

Theorem 3.1.

The asset market game G = G[(Wⁱ), (E_k, S_k)] has a Nash equilibrium.

At any equilibrium, all markets are active.

Every equilibrium is strict.

The proof of the theorem is essentially routine, based on the observation that the payoff functions are concave. Some care must be taken because of possible discontinuities at the boundary of the budget sets. Details are in the appendix.

The results on the structure of equilibrium in the next section are based on the following observation. The marginal return to investor i from asset k can be written as (12) ∂ π k i ( w ) ∂ w k i = r k [ 1 − w k i w k 1 1 + η k ] for w k > 0 where r_k = R_k(w_k), η_k = H_k[P_k(w_k)]. Indeed, by definition, x k i ( w ) = w k i / P k ( w k ), and by (4), 1 / 1 + η k = P k ′ w k / P k, therefore ∂ x k i ( w ) ∂ w k i = P k − w k i P k ′ ( P k ) 2 = 1 P k [ 1 − w k i w k w k P k ′ P k ] = 1 P k [ 1 − w k i w k 1 1 + η k ] Formula (12) follows immediately from the definitions (8) and r_k = E_k/P_k. The “Nash term” − w k i / w k ( 1 + η k ) in (12) reflects the fact that an increase of w k i reduces the return rate of asset k; it disappears only under the “competitive” assumption of infinitely elastic supply (η_k = ∞).

4. Structure of Nash Equilibrium

This section contains the main results. Consider an equilibrium w = (w¹,…w^N) of G with associated prices p_k = P_k(w_k), supplies S_k(p_k), asset returns r_k = R_k(w_k) = E_k/p_k, and elasticities η_k = H_k(p_k). Denote by E = E(w) = Σ_kE_kS_k(p_k) the aggregate payoff in the economy, and let R = E(w)/W be the aggregate rate of return (remember that W = Σ_iWⁱ is the aggregate initial wealth). If w k i > 0 we say that investor i holds asset k, or that he is active in market k. Denote further by r i : = π ¯ i ( α ) = π i ( w ) / W i the rate of return investor i gets on his capital, and write α i = ( α 1 i , … α K i ), where α k i = w k i / W i, for the portfolio associated with wⁱ.

Theorem 4.1.

Let w = ( w k i ) be an equilibrium, with investors and assets ordered such that W¹ ≥ W² ≥…≥ W^N and r₁ ≤ r₂ ≤…≤ r_K. Then

the largest investor (i = 1) holds all assets: w k 1 > 0 ∀ k,

the asset with the highest return (k = K) is held by every investor: w K i > 0 ∀ i,

if investor i holds asset k ( w K i > 0 ), then:

i also holds all assets with higher or equal returns ( w l i > 0 for r l ≥ r k )

all larger investors j ≤ i also hold at least the same quantity of asset k ( w k j ≥ w k i ), with strict inequality iff j is strictly larger than i (W^j > Wⁱ)

larger investors hold relatively more low-yielding assets in the following sense: whenever Wⁱ ≥ W^j, then the portfolios αⁱ, α^j satisfy α 1 i + α 2 i + … α k i ≥ α 1 j + α 2 j + … α k j ∀ k

the lower the elasticity of supply for an asset, the higher its return: r k < r l ⇔ η k > η l and r k = r l ⇔ η k = η l

larger investors have lower return rates: rⁱ ≥ r^j ⇔ Wⁱ ≤ W^j

Let w be an equilibrium, and denote by λⁱ the Lagrange multiplier associated with investor i's budget constraint. By formula (12), the following first-order conditions [FOC] must hold, for i = 1,…N: (13) ∂ π i ( w ) ∂ w k i = { r k [ 1 − w k i w k 1 1 + η k ] = λ i ∀ k with w k i > 0 r k ≤ λ i ∀ k with w k i = 0

The proof of the various assertions in the Theorem is based on a careful examination of these first-order conditions. Details are in the appendix.

Table 1 summarizes the results of Theorem 4.1. At a Nash equilibrium, in general, different assets have different returns; and not every investor is active in all markets. Larger investors are active in more markets. The more elastic the supply of an asset, the lower its return rate at equilibrium. Larger investors hold relatively more low-yielding assets, and achieve lower average rates of return on their capital. Investors with the same wealth use the same strategy.

Intuitively, r_k is the average return of asset k, and by (12), ∂ π i ( w ) ∂ w k i = r k [ 1 − w k i w k 1 1 + η k ] is the marginal return of asset k for investor i. The marginal return is always less than the average return r_k (because an extra dollar invested in an asset also pushes up its price), but it gets closer to r_k when the elasticity η_k increases. Since marginal, not average, returns must be equal at equilibrium, we get the inverse relationship between r_k and η_k asserted in the Theorem. Moreover, the discrepancy between marginal and average return increases with w k i, i.e. it is larger for larger investors

Thus with variable supply, Nash equilibrium allocations are not competitive in general (prices are not fair). Example 1 illustrates such a case. This deviation of asset prices from the expected return has nothing to do with risk aversion of our investors, but results from their strategic interaction in a situation where the supply conditions of different assets differ. Of course, in our model, for any asset k, the exogenous supply function S_k(p_k) summarizes the aggregate behavior of the (non-strategic) “rest of the market”. This “rest” may contain risk-averse traders (or even traders with no rational attitude to risk at all). While we do not model these traders explicitly, it may of course be that the elasticity of supply of some asset k depends on its riskiness; and to the extent that this is the case, our equilibrium prices also reflect risk, at least indirectly.

Moreover, we observe what we have termed the “curse of size”: larger investors achieve lower average return rates at equilibrium. Again this has nothing to do with any differences in the skills or preferences of investors, but results from the equalization of marginal, not average, return rates at a Nash equilibrium. A typical small investor concentrates his portfolio on the highest-yielding assets, achieving a high average return rate; and because he is small, his marginal return is high, too. A large investor has a much lower marginal return and finds it profit-maximizing to hold also the lower-yielding assets, thus depressing his average return.

Example 1

Let N = 2, K = 2, E_k = 1 ∀k, and S₁(p₁) = p₁, S₂(p₂) = 1. Then η₁ = 1, P 1 ( w 1 ) = w 1, R 1 ( w 1 ) = 1 / w 1 and η₂ = 0, P₂(w₂) = w₂, R₂(w₂) = 1/(w₂). Assume that the initial endowments are W¹ = 4.75, W² = does not hold asset, so that W = 5. Then the unique Nash equilibrium is given in Table 2. It is easy to check that the first-order conditions are satisfied, with r¹ > λ², i.e. investor i = 2 does not hold asset k = 1 ( w 1 2 = 0). The total payoff at equilibrium is E = π¹ + π² = 3, the average return is R = E/W = 0.6, and the competitive rate is r̂ = 0.558.

Theorem 4.2 below shows that the deviation of Nash equilibrium prices from their fair values is not due to the variability (as opposed to constancy) of supply per se, but to differences in the supply conditions of different assets. As a preliminary step, the following proposition shows that the only competitive profile that can possibly be a Nash equilibrium is the “proportional competitive” profile ŵ defined in Lemma 3.2.

Proposition 4.1.

Let w be a Nash equilibrium profile. Then w is competitive if and only if w = ŵ (i.e., all investors use the proportional investment rule, cf. Lemma 3.2)

Proof

If the equilibrium satisfies w = ŵ, it is competitive by Lemma 3.2 (i). Conversely, assume that a Nash equilibrium w is competitive. By Lemma 3.1, r_k = r̂, w_k = ŵ_k ∀k. By Theorem 4.1, all elasticities are equal, η_k = η̂ ∀k, and every investor i holds all assets. Therefore the first-order condition for an investor i takes the form r ^ [ 1 − w k i w ^ k 1 1 + η ^ ] = λ i ∀ k Thus, there is γⁱ> 0 such that w k i / w ^ k = γ i ∀ k, and all investors hold the same portfolio in percentage terms, αⁱ = α^j. By Lemma 3.2 (ii), w = ŵ. In particular, there is at most one competitive Nash equilibrium. A sufficient condition for the equilibrium to be competitive is given in the next theorem.

Theorem 4.2.

Assume that there exists a common elasticity function H(·) such that H_k(p_k) = H(p_k/E_k) for all assets k. Then there exists a unique equilibrium w, and w = ŵ.

The assumption of the Theorem means that all supply functions S_k(·) have the same elasticity function, provided units are chosen such that the payoff per unit is the same for all assets. Such a normalization (e.g., E_k = 1) is always possible w.l.o.g. (cf. footnote 8). In particular, the assumption of the theorem is satisfied (with H ≡ 0) if supply is constant.

Proof

We prove that all r_k are equal at equilibrium. Let r₁ ≤ r₂ ≤ … ≤ r_K as in Theorem 4.1. Then H₁(p₁) ≥ H_K(p_K), hence by assumption H(p₁/E₁) ≥H(p_K/E_K). By S.3, the function H is non-increasing, hence p₁/E₁ ≤ p_K/E_K or r₁ = E₁/p₁ ≥ E_K/p_K = r_K. Therefore r_k = r̂ ∀k and the equilibrium is competitive. By Proposition 4.1, w = ŵ uniquely.

A competitive equilibrium, if it exists, is symmetric. There may exist non-competitive symmetric equilibria (cf. Theorem 4.3), but a game can have at most one symmetric equilibrium. In fact, more is true: a game can have at most one equilibrium in which all investors are active in all markets:

Proposition 4.2.

There is at most one Nash equilibrium in which every investor holds all assets, w k i > 0 ∀ i , ∀ k.

Since in a symmetric equilibrium every investor must hold all assets, we obtain immediately:

Corollary 4.1.

There exists at most one symmetric Nash equilibrium.

Another interesting special case is when all investors have the same wealth, Wⁱ = W⁰∀i (but supply elasticities may differ).

Theorem 4.3.

Assume that all investors have the same wealth, Wⁱ = W⁰ > 0 ∀i. Then there exists a unique equilibrium, and all investors choose the same strategy: wⁱ = w^j ∀i,j.

Proof

Consider an equilibrium and number investors and assets as in Theorem 4.1. By assumption, all investors have the same wealth, and by monotonicity (45) w k i ≥ w k i + 1. This is only possible if w k i = w k j ∀ i , j, i.e., if w k i = 1 N w k ∀ i , k. Thus the equilibrium is symmetric, and by Corollary 4.1, unique.

Remark

If the Nash equilibrium is competitive, then all investors necessarily choose the same portfolio, by Proposition 4.1. The converse is not true: In Theorem 4.3, for example, all investors choose the same portfolio, but assets with constant, but different supply elasticities have different return rates.

Are the investors better off at Nash equilibrium than at a competitive profile? Consider an arbitrary profile w ∈ B in which all markets are active, so that the return rates r_k = r_k(w_k) are well defined for all k. Then the payoff of investor i can be written π i ( w ) = ∑ k w k i r k, and his (gross) rate of return (per dollar invested) is (14) r i = π ¯ i ( α ) = ∑ k w k i W i r k = ∑ k α k i r k a convex combination of the quantities r₁,…r_K. In a competitive profile, r_k = r̂ ∀k, so that of course rⁱ = r̂. If the profile w is not competitive, then some r_k must be strictly smaller than r̂, and some strictly larger (because the functions r_k = R_k (w_k) are strictly decreasing, and the sum Σ_kw_k = W is fixed). Thus it is not clear a priori if an investor's equilibrium rate of return is greater or smaller than r̂, especially it is not clear for large investors who hold relatively more low-yielding assets (Theorem 4.1 (4)). In fact, investors never do worse at a Nash equilibrium than at a competitive allocation:

Theorem 4.4.

Let w* ∈ B be a Nash equilibrium that is not a competitive allocation. Then every investor i achieves a strictly higher rate of return than the competitive rate r i = π i ( w ∗ ) W i > r ^ ∀ i

5. Evolutionarily Stable Strategies

The concept of an evolutionarily stable strategy (ESS) for a finite game introduced by [11] is defined for symmetric games as follows. A strategy s in the common strategy space S is an ESS if, starting from a symmetric situation where everybody uses the strategy s, the payoff of a single deviator after deviation is never greater than the payoff of the others (the non-deviators) after this deviation. I.e. no single deviation from the ESS improves the deviator's relative position.

Although this is a static concept, it can sometimes be shown that an ESS is also a stable rest point of some suitably specified “evolutionary” dynamic process of imitation and experimentation ([1]).

Since the game G is not symmetric due to the unequal wealth of different investors, neither the definition of an ESS nor the idea of imitation is directly applicable. But one can argue that these concepts make sense if we think in percentage terms, i.e., in the more symmetric formulation Ḡ. Indeed, in Ḡ every investor, large or small, has the same strategy space ∆^K, and the payoffs of different players can meaningfully be compared. An investor making 3% with a portfolio of a certain composition might look at some other investor (bigger or smaller) making 4% with a portfolio of a different composition, and might imitate the composition of the other, seemingly more successful, portfolio. In the spirit of bounded rationality, such imitative behavior is certainly justifiable. If we accept this, it becomes meaningful to define an ESS as a strategy which, if adopted by all, cannot be destabilised by a single deviation (under a dynamic driven by imitation of more successful players).

The game Ḡ resembles a symmetric game because all players have the same strategy space, and any two players using the same strategy αⁱ = α^j necessarily have the same payoff. But the game Ḡ is still not a symmetric game in the strict sense: if a large and a small investor with different strategies αⁱ, α^j interchange their strategies, this may change prices and hence may change the other players' payoffs. Nevertheless, as argued above, we may define a concept of ESS in Ḡ.

For a strategy α⁰ ∈ ∆^K, we denote by α → 0 = ( α 0 , … α 0 ∈ ( Δ K ) N the symmetric profile in which every player uses α⁰. Let us call a strategy α⁰ ∈ ∆^K an ESS of Ḡ if for every player i and for every strategy αⁱ ∈ ∆^K the following is true: π ¯ i ( ( α 0 | i α i ) ) ≤ π ¯ j ( ( α 0 | i α i ) ) ∀ j ≠ i where (α⁰ |_i αⁱ) denotes the strategy profile in which player i uses strategy αⁱ and every other player uses the strategy α⁰. That is, if a player deviates from the symmetric profile α → 0, then, after the deviation, his payoff is not larger than the payoff of any other player, so that nobody has an incentive to imitate the deviator. On the contrary, the deviator will have an incentive (at least in the weak inequality sense) to imitate one of the other players, i.e. to switch back to the ESS strategy α⁰.

Theorem 5.1.

The game Ḡ has a unique ESS, namely α⁰ = α̂, where α̂ is the competitive market portfolio corresponding to the “proportional competitive profile” ŵ defined after Lemma 3.1. At this ESS, all players have the same payoff in Ḡ, namely the competitive return π ¯ i ( α → 0 ) = r ^ ∀ i

Thus the ESS outcome is competitive, but different from the Nash outcome in general. Such a relation between ESS and competitive outcomes has been observed in other contexts as well, cf. [8,9].

Tables

Table 1.

The structure of equilibrium.

		η₁	≥	η₂	≥	…	≥	η_k	≥	…	≥	η_K
	w k i	r₁	≤	r₂	≤	…	≤	r_k	≤	…	≤	r_K	row sums
														r¹	λ¹	w 1 1	w 2 1	…	w k 1	…	w k 1	W¹
≤	≤							≥				≥	≥
r²	λ²	0	…		w k 2 2	…		w k 2		…		w K 2	W²
≤	≤							≥				≥	≥
…	…					…				…			…
rⁱ	λⁱ	0		0		…		w k i i		…		w K i	Wⁱ
≤	≤									…		≥	≥
…	…					…				…			…
r^N	λ^N	0		0		…		0	…	w k N N		w K N	W^N

	column sums	w₁		w₂		…		w_k		…		w^K	W

An equilibrium allocation: rows correspond to investors and columns to assets. w K i is the amount invested in asset k by investor i. If Wⁱ = W^j, then the corresponding rows are identical; if Wⁱ > W^j, then rⁱ ≤ r^j, λⁱ < λ^j, and w K i > w K j, except when w K i = 0. For any two adjacent columns k and k + 1, r_k = r_k₊₁ iff η_k = η_k₊₁. In this case, w K i > 0 ⇒ w K + 1 i > 0 ∀ i.

Table 2.

Nash equilibrium in Example 1.

		η₁ = 1	η₂ = 0
		r₁ = 0.5	r₂ = 1

r¹ = .578	λ¹ = 0.25	w 1 1	w 2 1	W¹ = 4.75	π¹ = 2.75
r² = 1	λ² = 0.75	w 1 2	w 2 2	W² = 0.25	π² = 0.25

R = 0.6		w₁ = 4	w₂ = 1	W = 5	E = 3

I thank C. Alós - Ferrer, A. Ania, K. Podczeck and A. Ramsauer for helpful conversations.

A. appendix Lemma A.1

Under assumption S.1

for every w_k ≥ 0, Equation (2) determines a unique price p_k = P_k(w_k). The function P_k(w_k) is continuous and strictly increasing on [0, ∞), with P k ( 0 ) = 0 , lim w k → ∞ P k ( w k ) = ∞

The function f_k(w_k) := w_k/P_k(w_k) is nondecreasing for w_k > 0 and (15) lim w k → 0 w k / P k ( w k ) = S k ( 0 )

Proof

define the function (16) V k ( p k ) = p k S k ( p k ) for p k ≥ 0 By S.1, V_k(p_k) is continuous and strictly increasing on [0, ∞), with V_k(0) = 0 and lim p k → ∞ V k ( p k ) = ∞. Therefore V_k has an inverse V k − 1 with the same properties. Since equation (2) can be written V_k(p_k) = w_k the price function is equal to this inverse, P k ( w k ) = V k − 1 ( w k ).

For w_k > 0, also P_k(w_k) > 0, and f_k(w_k) = w_k/P_k(w_k) ≡ S_k(P_k(w_k)). The assertion follows from S.1 and (i).

Remark

Conversely, the properties of the price function P_k stated in Lemma A.1 imply that the supply function S_k satisfies S.1. Indeed, if we postulate an arbitrary price function P_k with the properties stated in Lemma A.1, and define a supply function S_k by the condition P_k(w_k)S_k(P_k(w_k)) ≡ w_k for w_k > 0, and by (15) for w_k = 0, then S_k satisfies S.1. To see this, write S_k(P_k(w_k)) = w_k/P_k(w_k) and observe that the 1-1-transformation w_k ↔ p_k = P_k(w_k) is strictly increasing.

For future reference, we note that for any c > 0 (17) S k ( P k ( ε ) ) > ε c ∀ ε > 0 sufficiently small

(Since S_k(P_k(ε))/ε = 1/P_k(ε)).

The following is a more precise statement of the differentiability assumption in S.1. It is phrased so that an infinite slope at p_k = 0 is not excluded.

S.4

For all k, the supply function S_k(p_k) is either

twice continuously differentiable on [0, ∞) with S k ′ ( 0 ) finite, or

twice continuously differentiable on (0, ∞), with lim h → 0 ( S ( h ) − S ( 0 ) ) / h = S k ′ ( 0 ) = ∞ = lim p k → 0 S k ′ ( p k )

¿From now on, we maintain the assumptions S.1, S.4, S.3. Clearly, the elasticity function η k = H k ( p k ) = p k S k ′ ( p k ) / S k ( p k ) is continuously differentiable on (0, ∞) and by S.3, the limit lim p k → 0 H k ( p k ) = : H k ( 0 ) ∈ [ 0 , ∞ ] exists (possibly infinite).

Lemma A.2.

There are only two possible cases: either

supply is constant, H_k(0) = 0, and S_k(p_k) = S_k(0) = S̄_k > 0 ∀p_k ≥ 0, or

supply is not constant, H_k(0) > 0, and S_k(p_k) > S_k(0) = 0 ∀p_k > 0; moreover

(α) S k ′ ( 0 ) = 0 if H_k(0) > 1 (supply is elastic at 0)

(β) S k ′ ( 0 ) = ∞ if 0 < H_k(0) < 1 (supply is inelastic at 0)

(γ) if H_k(0) = 1, it may be that S k ′ ( 0 ) is positive and finite. (iii) in any case, lim p k → 0 p k S k ′ ( p k ) = 0

Proof

We omit the subscript k for simplicity.

Clearly, H(0) = 0 iff H(p) = 0 ∀p > 0, i.e. iff S′(p) = 0 ∀p > 0, i.e. iff supply is constant (and positive, by S.1): S(p) = S(0) > 0 ∀p ≥ 0. Obviously (iii) is satisfied in this case.

Assume now that supply is not constant. Then H(0) > 0. By S.1 S(p) > 0 for p > 0. Next we show that S(0) = 0. By definition, (18) p S ′ ( p ) ≡ H ( p ) S ( p ) ∀ p > 0 and (19) d d p S ( p ) p = p S ′ ( p ) − S ( p ) p 2 ∀ p > 0

Consider first the case of elastic supply at 0, i.e. H(0) > 1. Then, for p > 0 sufficiently small, H(p) > 1, and (18) implies: pS′(p) > S(p), i.e., by (19), the positive function S(p)/p is strictly increasing in p. Therefore lim p → 0 ( S ( p ) / p ) exists and is nonnegative and finite. This implies that S(0) = 0, and furthermore that S ′ ( 0 ) = lim p → 0 ( S ( p ) − S ( 0 ) ) / p = lim p → 0 S ( p ) / p is finite. Moreover by (18): S ′ ( p ) ≡ H ( p ) S ( p ) p

Both S′(p) and S(p)/p tend to the same finite limit S′(0) as p → 0, whereas H(p) is bounded away from 1 for all p sufficiently small. This is possible only if S′(0) = 0. This proves (ii)(α). Clearly (iii) is also satisfied in this case.

Consider now the case of inelastic or unit elastic supply at 0, 0 < H(0) ≤ 1. Then, for p > 0, H(p) ≤ 1 by S.3, and (18) implies: pS′(p) ≤ S(p), i.e., by (19), the positive function S(p)/p is (weakly) decreasing in p. Therefore lim p → 0 ( S ( p ) / p ) exists and is strictly positive (possibly infinite).

Since H ( p ) = S ′ ( p ) S ( p ) / p is also weakly decreasing by S.3, the function S′(p) must be weakly decreasing, i.e. the supply function S(p) is concave. This implies (20) S ′ ( p ) ≤ S ( p ) − S ( 0 ) p ∀ p > 0

It implies also that S′(0) > 0 (possibly S′(0) = ∞), since otherwise S′(p) ≡ 0 and supply would be constant.

We want to show that S(0) = 0. If S(0) > 0, then, for p > 0 sufficiently small, S ( p ) [ 1 − H ( 0 ) 2 ] < S ( 0 )because S(.) is continuous and 0 < H(0) ≤ 1. Therefore S ( p ) − S ( 0 ) < 1 2 H ( 0 ) S ( p ) and, by (20) (21) S ′ ( p ) < H ( 0 ) 2 S ( p ) p for p sufficiently small .

On the other hand, by (18), S ′ ( p ) ≡ H ( p ) S ( p ) p and or p sufficiently small: H ( p ) > 1 2 H ( 0 ) (because H(0) > 0. This implies S ′ ( p ) > H ( 0 ) 2 S ( p ) p for p sufficiently small , contradicting (21). Therefore S(0) = 0 for 0 < H(0) ≤ 1 as well. Using (18) we see that lim p → 0 p S ′ ( p ) = lim p → 0 H ( p ) S ( p ) = H ( 0 ) S ( 0 ) = 0 so that (iii) is also satisfied.

Finally, since S(0) = 0, we have S ′ ( 0 ) = lim p → 0 ( S ( p / p ) ), and using (18) again: S ′ ( p ) = H ( p ) S ( p ) P

If p → 0, both S′(p) and S(p)/p tend to the same positive limit S′(0) (possibly infinite) and H(p) tends to H(0). If 0 < H(0) < 1 this is possible only if S′(0) = ∞. This proves (ii)(β). If H(0) = 1, it is possible that S′(0) is positive and finite; e.g., for S(p) = p, S′(p) = 1, H(p) = 1 ∀p ≥ 0. This proves (ii)(γ) and the Lemma.

Lemma A.3

The function V_k(p_k) = p_kS_k(p_k) is continuously differentiable on [0,∞), with V k ′ ( p k ) > 0 for p_k > 0 and V k ′ ( 0 ) = S k ( 0 ) .

The price function P_k(w_k) is continuously differentiable on [0, ∞) [resp. on (0, ∞)], if S_k(0) > 0 [resp. S_k(0) = 0]; with P k ′ ( w k ) > 0 for w_k > 0 and (22) P k ′ ( 0 ) = 1 S k ( 0 ) = lim w k → 0 P k ′ ( w k ) (where 1/S_k(0) = ∞ if S_k(0) = 0).

proof

For p_k > 0, the assertions are trivial. At p_k = 0, we have: V k ′ ( 0 ) = lim ε → 0 ( V k ( ε ) − V k ( 0 ) ) / ε = lim ε → 0 ( ε S k ( ε ) − 0 ) / ε = S k ( 0 )

For p_k > 0, we have: V k ′ ( p k ) = S k ( p k ) + p k S k ′ ( p k )

By Lemma A.2 (iii), the last term goes to zero for p_k→ 0, hence lim p k → 0 V k ′ ( p k ) = lim p k → 0 S k ( p k ) = S k ( 0 )

This proves (i).

The price function P_k is the inverse of the function V_k. The assertions follow immediately from (i) and this fact, noting that P k ′ ( w k ) = 1 / V k ′ ( P k ) at all points where V k ′ is positive, and that P_k has infinite slope at zero iff V k ′ ( 0 ) = 0 .

Lemma A.4.

The game G is constant-sum (on the set {w ∈ B | w_k > 0 ∀k} of strategies where all markets are active) if and only if all supply functions S_k(·) are constant

Proof

The “if” part is trivial. Assume now that the game is constant-sum, i.e., ∑ i π i ( w ) = ∑ i ∑ k w k i R k ( w k ) = ∑ k w k R k ( w k ) = ∑ k f k ( w k ) = const . for all w_k > 0 with Σ_kw_k = W. This implies f k ′ ( w k ) = f l ′ ( w l ) = c ∀ w k , w l, and f_k(w_k) = cw_k + d_k ∀k, for some constants c ≥ 0, d_k ≥ 0. If c > 0, then S_k(P_k(w_k)) = f_k(w_k) = cw_k + d_k is not constant, hence lim p k → 0 S k ( p k ) = 0 by Lemma A.2, hence d k = lim w k → 0 f k ( w k ) = 0. But then f_k(w_k) = cw_k = w_kR_k(w_k) ⇒ R_k(w_k) = c, contradicting Lemma A.1. Therefore c = 0 and S_k[P_k(w_k)] = f_k(w_k) = d_k > 0, i.e. supply S_k is constant.

Proof of Lemma 3.2

Assume that all investors follow the proportional rule. Then (23) w k i = γ i E k S k ∀ i , k

Summing this over i gives w k = ( ∑ i γ i ) E k S k ⇔ R k = ( E k S k ) / w k = ( ∑ i γ i ) − 1 = : r ̂, i.e. the profile is competitive. Thus ŵ_k is uniquely determined by R_k(ŵ_k) = r̂. Hence E_kS_k = r̂Ŵ_k. Summing this over k gives E(w) = r̂W, and summing (23) over k gives Wⁱ = γⁱE(w) ⇔ γⁱ = Wⁱ/r̂W. Therefore w k i = γ i E k S k = ( W i / r ̂ W ) r ̂ W = w ̂ k W / W i. This proves (i).

Since w is competitive, w_k = ŵ_k where R_k(ŵ_k) = r̂ ∀k. Since all agents hold the same portfolio αⁱ = α^j, summing w k i = α k i w i over i gives w ̂ k = α k i W ⇔ α k i = w ̂ k / W ⇒ w k i = α k i W i = w ̂ k W i / W. This is the allocation ŵ given in part (i) of the lemma.

To prepare for the proof of Theorem 3.1, note that the payoff functions πⁱ(w) are defined for all nonnegative vectors w ∈ ℝ + K N, independently of the agents' budget constraints. Clearly, the functions x k i ( w ) and also the payoff functions πⁱ(w) are differentiable in w k i at all points where w_k> 0 (with one-sided derivatives if w k i = 0, but w_k> 0). Denote by W a = { w ∈ ℝ + K N | w k > 0 ∀ k } the set of profiles where all markets are active. Note that W^a is convex and the payoff functions πⁱ(w) are continuous and differentiable on W^a.

First we compute some derivatives. Let w be a profile at which market k is active, i.e., w_k> 0, p_k> 0, R_k> 0. We have (24) x k i ( w ) = w k i P k ( w k ) = w k i w k S k ( P k ( w k ) )

Therefore, from (12), (25) ∂ x k i ( w ) ∂ w k i = 1 P k ( w k ) [ 1 − w k i w k 1 1 + H k [ P k ( w k ) ] ] ≥ 0 with strict inequality unless investor i is the only one who buys asset k ( w k i = w k ) and the supply elasticity is zero (η_k = H_k[P_k(w_k)] = 0). Also (26) ∂ x k i ( w ) ∂ w k i ≤ 1 P k with strict inequality unless w k i = 0, and (27) ∂ x k i ( w ) ∂ w k i → ∞ for w k → 0 provided the expression [ 1 − w k i w k 1 1 + η k ] remains bounded away from 0 as w_k → 0 (this is certainly the case if w k i / w k remains bounded away from 1). Moreover (28) ∂ 2 x k i ( w ) ∂ ( w k i ) 2 = − P k ′ [ P k ] 2 [ 1 − w k i w k 1 1 + H k ] − 1 P k [ w k − w k i ( w k ) 2 1 1 + H k + w k i w k − H k ′ P k ′ ( 1 + H k ) 2 ] ≤ 0 where P_k = P_k(w_k) and H_k = H_k[P_k(w_k)]. The inequality follows because η k ′ = H k ′ ( p k ) ≤ 0 by S.3, and is strict unless w k i = w k and both η_k = 0 and η k ′ = 010. The cross-partials are (29) ∂ 2 x k i ( w ) ∂ w k i ∂ w l i = 0 for l ≠ k .

By (8) similar formulae hold for the profit functions πⁱ, e.g., (30) ∂ π i ( w ) ∂ w k i = E k ∂ x k i ( w ) ∂ w k i ≥ 0 for w k > 0

The formal proof of Theorem 3.1 is preceded by some lemmas.

Lemma A.5.

For all i, the payoff function of investor i, πⁱ(w) = πⁱ(wⁱ, w⁻ⁱ) is concave in i's own strategy wⁱ on the set W^a, and even strictly concave except possibly at points where w k i = w k for some k (investor i is the only buyer of asset k).

Proof

We have, on the convex set W^a (31) ∂ 2 π i ( w ) ∂ ( w k i ) 2 = E k ∂ 2 x k i ( w ) ∂ ( w k i ) 2 ≤ 0 with strict inequality for w k i < w k and all cross-partials are zero.

Lemma A.6.

Let w̄ = (w̄¹,…w̄^N) ∈ B be a strategy profile at which not all markets are active (w̄ ∉ W^a). Then every investor has a profitable deviation, i.e. for every i there exists a ŵⁱ ∈ Bⁱ such that (32) π i ( w ̂ i , w ̄ − i ) > π i ( w ̄ )

Moreover, ŵⁱ can be chosen so that at the new profile ŵ = (ŵⁱ, w̄⁻ⁱ) all markets are active.

Proof

Fix an investor i. Since he must invest his wealth somewhere, there exists an asset m such that w ̄ m i > 0 ( ⇒ w ̄ m > 0 , p ̄ m > 0 , R ̄ m > 0 ). Let l be an inactive asset so that w ̄ l = w ̄ l i = 0. Consider the following change in i's strategy, for small ε > 0: w ̂ m i = w ̄ m i − ε , w ̂ l i = ε , w ̂ k i = w ̄ k i for k ≠ m , l .

That is, investor i shifts a small amount ε from asset m to the inactive asset l. This shift decreases his earnings in market m by (using (26), (30)) 0 ≤ ε ∂ π i ( w ̄ ) ∂ w m i ≤ ε E m 1 p ̄ m = ε R ̄ m and it increases his earnings in market l by E_lS_l(P_l(ε)). By (17) this is strictly greater than εR̄_m for ε sufficiently small, i.e. (32) is satisfied. If l is the only inactive market at w̄, we are done. If not, repeat the construction for the next inactive asset, starting from the profile ŵ = (ŵⁱ, w̄⁻ⁱ).

As an immediate Corollary we have that all markets must be active at equilibrium.

Lemma A.7

If w is an equilibrium, then (33) ∂ π i ( w ) ∂ ( w k i ) > 0 and ∂ 2 π i ( w ) ∂ ( w k i ) 2 < 0 for all i = 1, … N, k = 1, … K.

Proof

By Lemma A.6, w_k > 0 at equilibrium. Therefore, by (25), (28), both claims are true unless (34) w k i = w k and η k = 0 ( and η ′ k = 0 ) .

We have to show that this situation is impossible at equilibrium.

Indeed, if (34) holds, then w k i = w k > 0 and ∂ π i ( w ) / ∂ w k i = 0 by (25).

On the other hand. there must exist an asset l with w l i < w l, hence by (25) ∂ π i ( w ) / ∂ w l i > 0

Shifting a small amount ε > 0 from asset k to asset l increases i's profits, contradicting equilibrium.

In particular, if an asset is in constant supply, then it must be held by more than one investor at equilibrium. Moreover, if w* = (w*ⁱ, w*⁻ⁱ) is an equilibrium, then the payoff function π(wⁱ, w*⁻ⁱ) is strictly concave in wⁱ in a neighborhood of w*ⁱ, and concave elsewhere. Therefore every equilibrium is strict.

Proof of Theorem 3.1

Assertions (ii) and (iii) follow from the two preceding Lemmas. It only remains to prove assertion (i) (existence of Nash equilibrium).

For ν = 1, 2, 3, … consider the modified game G^ν with budget sets B i ( ν ) = { w i ∈ B i | w k i ≥ 1 ν ∀ k } and payoff functions πⁱ as before. Eventually, for ν sufficiently large, Bⁱ(ν) is nonempty, compact, convex. Clearly at any w ∈ B(ν) = Π_i Bⁱ(ν) ⊂ W^a all markets are active and each payoff function πⁱ(w) = πⁱ(wⁱ, w⁻ⁱ) is continuous in w ∈ B(ν) and strictly concave in the own strategy wⁱ. Therefore there exists an equilibrium w(ν) = (w¹(ν), … w^N(ν)) of the modified game G^ν, where of course w k i ( ν ) ≥ 1 ν always.

W.l.o.g. (passing to a subsequence if necessary) we may assume that the sequence w(ν)_ν_=1,2,… converges, i.e., w ( ν ) → w ∗ = ( w ∗ 1 , … w ∗ N ) for ν → ∞

We claim that w^* is an equilibrium in the unrestricted game G with strategy spaces Bⁱ. We proceed in two steps:

Step 1. w k ∗ > 0 ∀ k, i.e. w* ∈ W^a

Step 2. ∀i, w*ⁱ is a best reply to w*

Step 1

Assume, indirectly, that there is an asset l with w l ∗ = 0, i.e., w_l(ν) → 0 for ν → ∞. Of course then also w l i ( ν ) → 0 for each agent i; but since ∑ i w l i ( ν ) = w l ( ν ) > 0 always, there exists an agent j such that w l j ( ν ) / w l ( ν ) remains bounded away from 1 as ν → ∞ (taking a further subsequence if necessary). By (27) this implies ∂ x l j ( w ( ν ) ) ∂ w l j → ∞ as ν → ∞

There must also exist some asset m ≠ l such that w m ∗ j > 0, and hence w m ∗ > 0. Fix j, l, m. Then ∂ x m j ( w ( ν ) ) ∂ w m j = 1 P m ( w m ( ν ) ) [ 1 − w m j ( ν ) w m ( ν ) 1 1 + H m ( P m ( w m ( ν ) ) ) ] converges to the finite number 1 P m ( w m ∗ ] [ 1 − w m ∗ j w m ∗ 1 1 + H m ( P m ( w m ∗ ) ) ] = : c m ≥ 0 as ν → ∞. Therefore, for ν sufficiently large, agent j can increase his payoff π^j(w(ν)) in the game G^ν by shifting a small amount ν > 0 away from asset m (this is feasible because eventually w m ∗ j > 1 ν) to asset l. This contradicts the assumption that w(ν) is an equilibrium in G^ν and proves Step 1.

Step 2

By Step 1, πⁱ(w) is continuous at w*. Fix an investor i. We have to show that w*ⁱ is a best reply to w*⁻ⁱ. Assume not. Then there exists ŵⁱ ∈ Bⁱ which is a better reply to w*⁻ⁱ, i.e. (35) π i ( w ̂ i , w ∗ − i ) − π i ( w ∗ ) > δ for some δ > 0

If the strategy profile (ŵⁱ, w*⁻ⁱ) ∉ W^a, then by Lemma A.6 there exists a further deviation ŵ̂ⁱ such that (ŵ̂ⁱ, w*⁻ⁱ) ∈ W^a and πⁱ(ŵⁱ, w*⁻ⁱ) > πⁱ(wⁱ, w^*⁻ⁱ). Therefore we may assume that (35) holds with ŵ: = (ŵⁱ, w*⁻ⁱ) ∈ W^a, so that πi (·) is continuous at this point. Approximate ŵⁱ ∈ Bⁱ by a sequence ŵⁱ(ν) ∈ Bⁱ(ν). Then by continuity π i ( w ̂ i ( ν ) , w − i ( ν ) ) − π i ( w ( ν ) ) > δ 2 > 0 for ν sufficiently large, i.e. w(ν) is not an equilibrium of G^ν, contrary to assumption. This proves Step 2 and completes the proof of Theorem 3.1.

Proof of Theorem 4.1

Consider a Nash equilibrium w = (w¹, … w^N) with associated prices p_k, asset returns r_k, and elasticities η_k. Denote by α i = ( α 1 i , … α K i ) = ( 1 / W i ) w i the equilibrium portfolio, and by rⁱ:= πⁱ(w)/Wⁱ the average return of investor i.

The following proof is based on a careful examination of the first-order conditions for a Nash equilibrium. To understand the following arguments, it helps to keep Table 1 in mind.

Proof of Assertions 4.1–2

Let w be a Nash equilibrium. By (13) (36) ∂ π i ( w ) ∂ w k i = r k [ 1 − w k i w k 1 1 + η k ]

Denote by λⁱ the Lagrange multiplier associated with investor i's budget constraint. Then the following first-order conditions [FOC] must hold, for i = 1,…N: (37) ∂ π i ( w ) ∂ w k i = { r k [ 1 − w k i w k 1 1 + η k ] = λ i ∀ k with w k i > 0 r k ≤ λ i ∀ k with w k i = 0

By Lemma A.7, λⁱ > 0 for ∀i. (13) implies, ∀i, ∀k: (38) λ i < r k ⇔ w k i > 0 ( investor i holds asset k ) λ i ≥ r k ⇔ w k i = 0 ( investor i does not hold asset k ) }

W.l.o.g., order the investors such that (39) λ 1 ≤ λ 2 ≤ ⋯ ≤ λ N and order the assets such that (40) r 1 ≤ r 2 ≤ ⋯ ≤ r K .

We shall see below (see (46)) that (39) implies W¹ > W² > … W^N, i.e., investor i = 1 is the largest and i = N is the smallest investor. Similarly, asset k = 1 is the worst and k = K is the best asset, where “better” assets have higher returns per dollar invested.

For given i, define (41) k i : = min { k | λ i < r k } ( worst asset held by i ) and for given k, define (42) i k : = max { i | λ i < r k } ( smallest investor holding k )

It is easy to see that investor i holds exactly the assets k = kⁱ,kⁱ + 1,… K and (43) 1 = k 1 ≤ k 2 ≤ ⋯ ≤ k N ≤ K with kⁱ = k^j if λⁱ = λ^j. Similarly, it is also easy to see that asset k is held exactly by the investors i = 1,2,…i_k and (44) 1 ≤ i 1 ≤ i 2 ≤ ⋯ ≤ i K = N with i_k = i_l if r_k = r_l.

The equilibrium allocation w = ( w k i ) is summarized in Table 1.

The largest investor i = 1 holds all assets ( w k 1 > 0 ∀ k ) and the best asset k = K is held by all investors ( w K i > 0 ∀ i ). For fixed k, we know from (13), (39) that (45) w k i ≥ w k i + 1 with strict inequality iff [ w k i > 0 and λⁱ < λⁱ⁺¹]. Therefore the row sums Wⁱ in Table 1 satisfy (46) W 1 ≥ W 2 ≥ ⋯ ≥ W N with strict inequality (47) W i > W i + 1 iff λ i < λ i + 1 (since w K i > 0 ∀ i). This proves the first three assertions in Theorem 4.1.

Proof of assertion 5

let r_k ≤ r_l. Then i_k ≤ i_l and w.l.o.g. k < l. Summing the first line in (13) for asset k over i = 1,…i_k gives (since ∑ i = 1 i k w k i = w k) (48) r k [ i k − 1 1 + η k ] = ∑ i = 1 i k λ i = : Λ

Summing the first line in (13) for asset l also over i = 1, … i_k gives (49) r l [ i k − 1 1 + η l ] = Λ if i l = i k (50) r l [ i k − ∑ i = 1 i k w l i w l 1 1 + η l ] = Λ if i l > i k

If i_l = i_k the assertion follows directly from (48), (49). If i_l > i_k then necessarily r_l > r_k (per def. of i_k), and (48), (50) imply i k − 1 1 + η k > i k − ∑ i = 1 i k w l i w l 1 1 + η l 1 1 + η k < ∑ i = 1 i k w l i w l 1 1 + η l ≤ 1 1 + η l ⇒ η_k > η_l, and assertion 5 is proved.

Proof of assertion 6

write r i = π i ( w ) / W i = ∑ k ( w k i / W i ) r k = ∑ k α k i r k, where α k i = w k i / W i is the portfolio associated with the equilibrium strategy wⁱ. By Abel's summation formula, we can write r i = ∑ k = 1 K α k i r k = A K i r K + 1 + ∑ k = 1 K A k i ( r k − r k + 1 ) where A k i = ∑ l = 1 K α l i and r_K+₁ is arbitrary.

Now fix two investors i ≥ j so that Wⁱ ≤ W^j (investor i is smaller). We want to show that rⁱ ≥ r^j. From Abel's formula, noting that A K i = A K j = 1: r i − r j = ∑ k = 1 K − 1 ( A k i − A k j ) ( r k − r k + 1 )

Since r_k ≤ r_k₊₁ it suffices to show that A k i ≤ A k j for k = 1,… K − 1,or equivalently, (51) ∑ l = k + 1 K α l i ≥ ∑ l = k + 1 K α l i for k = 1 , … K − 1 (because A l i = 1 − ∑ l = k + 1 K α l i). Thus assertion 6 follows from assertion 4.

Proof of assertion 4

Note that the coefficients α k i have also the “triangular structure” exhibited in Table 1: If some α k i > 0, then also α k + 1 i > 0 , α k j > 0 , α k + 1 j > 0 ; and r_k − λⁱ. > 0.

For α k i > 0 the first-order condition (13) can be written r k [ 1 − α k i W i w k 1 1 + η k ] = λ i ⇔ α k i = r k − λ i W i w k ( 1 + η k ) r k

This implies α k i α k + 1 i = w k ( 1 + η k ) r k r k − λ i W i w k + 1 ( 1 + η k + 1 ) r k + 1 r k + 1 − λ i W i and α k i / α k + 1 i α k j / α k + 1 j = ( r k − λ i ) / ( r k + 1 − λ i ) ( r k − λ j ) / ( r k + 1 − λ j ) = : B (all quantities are positive by the remark made above). From Table 1 we know that λⁱ ≥ λ^j, so that r_k ₊₁ ≥ r_k > λⁱ ≥ λ^j. But then the function f(λ) = (r_k − λ)/(r_k₊₁ − λ) is decreasing in λ, hence B ≤ 1. This implies (52) α k i α k + 1 i ≤ α k j α k + 1 j

It remains to show that this implies (51).

Let k₀ be the first (smallest) k such that α k i > 0. Then for k₀ ≤ k ≤ m ≤ K α k i = β k m i α m i where β k m i : = α k i α k + 1 i α k + 1 i α k + 2 i ⋯ α m − 1 i α m i for k < m and β m m i = 1

Defining β k m j similarly, we see from (52) that β k m i ≤ β k m j for k₀ ≤ k ≤ m ≤ K.

Claim 1

α K i ≥ α K j

Proof

assume the contrary, α K i < α K j. Then α K i = β k K i α K i < β k K i α K i = α k j for k₀ ≤ k ≤ K − 1, and ∑ k = 1 K α k i = ∑ k = k 0 K α k i < ∑ k = 1 K α k j. But this is impossible because both sums must be equal to one.

Claim 2

α K i + α K − 1 i ≥ α K j + α K − 1 j

Proof

assume the contrary, α K i + α K − 1 i < α K j + α K − 1 j. Then α K − 1 i < α K − 1 j by Claim 1. Therefore α k i = β k , K − 1 i α K − 1 i < β k , K − 1 j α K − 1 j = α k j for k₀ ≤ k ≤ K − 2. Again the same contradiction arises.

Claim 3

α K i + α K − 1 i + α K − 2 i ≥ α K j + α K − 1 j + α K − 2 j

Proof

as before, assuming the contrary implies α K − 2 i < α K − 2 j by Claim 2, and this implies that α k i < α k j for k₀ ≤ k ≤ K − 3, leading to a contradiction.

Proceeding in this manner until K − l = k₀, we obtain all the inequalities (51) (the remaining ones are trivial). This proves assertion 4 and completes the proof of Theorem 4.1.

Proof of Proposition 4.2

Let w be an equilibrium, with w k i > 0 ∀ i , ∀ k. Then the first-order conditions (13) take the form (remember r_k = E_k/p_k) (53) E k [ 1 − w k i w k 1 1 + η k ] = λ i p k ∀ i , k

Summing over i gives (54) E k [ N − 1 1 + η k ] = Λ p k ∀ k where Λ:= Σ_i λⁱ > 0. The RHS of this equation is strictly increasing in p_k, and the LHS is weakly decreasing in p_k by S.3, hence, for any given Λ, there exists only one solution pk. If Λ increases, the curve described by the RHS shifts upwards, and the LHS does not change, i.e., the intersection point with the LHS shifts to the left, i.e., p_k decreases, ∀k. Since the price function p_k = P_k(w_k) is strictly increasing, w k = P k − 1 ( p k ) also decreases strictly in Λ, ∀k. Since Σ_kw_k = Σ_iWⁱ = W is constant, the numbers w_k (and hence also p_k, η_k) are uniquely determined by (54) and the condition Σ_kw_k = W. Given this, the numbers w k i (and the multipliers λⁱ) are uniquely determined by (53) and the budget constraints ∑ k w k i = W i.

Proof of Theorem 4.4

Write r k ∗ = R k ( w k ∗ ) for the Nash quantities and r̂ = R_k(ŵ_k) for the competitive values. Clearly, r k ∗ {  } r ^ ⇔ w k ∗  } w ^ k

Fix an investor i. Denote by w k ∗ − i = w k ∗ − w k ∗ i the total amount of money invested in asset k by the ‘others’. Define the sets K 0 = { k | w ^ k > w k ∗ − i } , K 1 = { k | w ^ k = w k ∗ − i } , K 2 = { k | w ^ k < w k ∗ − i } . Then K₀ is nonempty (because ∑ k w ^ k = W > W − W i = ∑ k w k ∗ − i), and W i = W − ∑ k w k ∗ − i = ∑ k ( w ^ k − w ∗ − i ) = ∑ k ∈ K 0 ( w ^ k − w ∗ − i ) + ∑ k ∈ K 2 ( w ^ k − w ∗ − i )

If K₂ ≠ Ø, the last sum is negative, hence ∑ k ∈ K 0 ( w ^ k − w ∗ − i ) > W i. Therefore, investor i can find a deviating strategy wⁱ ∈ Bⁱ such that w k i = 0 ∀ k ∉ K 0 and w ^ k > w k ∗ − i + w k i ∀ k ∈ K 0. Then R k ( w k ∗ − i + w k i ) > r ^ for k ∈ K₀ and π i ( w i , w ∗ − i ) = ∑ k ∈ K 0 w k i R k ( w k ∗ − i + w k i ) > W i r ^, i.e. with the strategy wⁱ the investor achieves a rate of return higher than r̂. Since w^*i is a best reply to w^*⁻ⁱ, we have πⁱ(w^*) ≥ πⁱ(wⁱ,w^*⁻ⁱ) > r̂.

If K₂ = Ø, then w k ∗ − i ≤ w ^ k ∀ k and the strategy wⁱ given by w k i : = w ^ k − w k ∗ − i ∀ k is feasible for investor i. The profile (wⁱ,w^*−i) ∈ B is competitive and guarantees player i the return r̂. By assumption, the equilibrium w^* is not competitive, hence w^*i ≠ wⁱ. Moreover, since any equilibrium is strict by Theorem 3.1, we must have πⁱ(w^*) > πⁱ (wⁱ, w*⁻¹) = r̂.

Proof of Theorem 5.1.

By Lemma 3.2 (ii) there exists a unique, strictly positive α 0 = ( α 1 0 , … α K 0 ) ∈ Δ K such that the return r_k = R₀ = r̂ in all markets is the same, viz. α⁰ = α̂. We claim that this α̂ is ESS.

Fix an investor i and let him deviate to some strategy α⁰ + ε ∈ ∆^K, where ε = (ε₁,…ε_k) ≠ 0 and of course Σ_k ε_k = 0. Denote by α′ the new profile where player i uses strategy α⁰ + ε and all other players use strategy α⁰. Write R ₀ + dR_k for the return rate in market k after this deviation, i.e., at the profile α'.

The crucial observation is that ∀k ε_kdR_k < 0 if ε_k ≠ 0. Indeed, if ε_k> 0, then more money is invested in asset k, which strictly increases the price of this asset and strictly decreases the return R_k; and conversely for ε_k < 011.

The deviator's payoff after the deviation is π ¯ i ( α ′ ) = ∑ k ( α k 0 + ɛ k ) ( R 0 + d R k ) = ∑ k α k 0 ( R 0 + d R k ) + ∑ k ɛ k R 0 + ∑ k ɛ k d R k

The second term in the last expression is zero, and the third term is negative, by the observations made above. Therefore the payoff of a non-deviator j (j ≠ i) after the deviation is greater: π ¯ j ( α ′ ) = ∑ k α k 0 ( R 0 + d R k ) > π ¯ i ( α ′ )

This proves that α⁰ is an ESS. It remains to show uniqueness.

Let β⁰ ≠ α⁰ be any other strategy. We have to show that the symmetric profile β⃗⁰ = (β⁰, … β⁰) ∈ (∆^K)^N is not an ESS, i.e., we have to show that there exists a deviation ε = (ε₁,… ε_k) with Σ_kε_k = 0 which improves the deviator's relative position.

First it is clear that if β k 0 = 0 for some k, then the rate of return for sufficiently small investments in market k is arbitrarily large, and a small shift of money into market k helps the deviator more than the others. Assume therefore that β⁰ is strictly positive.

Denote by R_k the rate of return in market k under the symmetric profile β⃗⁰. Since β⁰ ≠ α⁰ we know from the first part of the proof that not all R_k are equal, i.e. the vector (R₁,… R_K) is not orthogonal to the hyperplane L = {z = (z₁, … z_K) ∈ ℝ^K| Σ_kz_k = 0} in ℝ^K. Therefore we can find a vector z ∈ L such that Σ_kz_kR_k > 0. Clearly, for t > 0 sufficiently small, the vectorε; = ε(t) = tz is a feasible deviation, i.e. β⁰ + ε(t) ∈ ∆^K.

Denote by R_k + dR_k the return rate in market k after such a deviation, i.e. at the new profile β′ = (β⁰ |_i β⁰ + ε). Then, for the deviator i and any non-deviator j: π ¯ i ( β ′ ) − π ¯ j ( β ′ ) = ∑ k ( β k 0 + ɛ k ) ( R k + d R k ) − ∑ k β k 0 ( R k + d R k ) = ∑ k ɛ k ( R k + d R k ) = ∑ k t z k ( R k + d R k ) = t ∑ k z k ( R k + d R k )

The last term is positive for t > 0 sufficiently small, because dR_k → 0 for t → 0. Thus we have found a deviation which makes the deviator better off than the others, and β⁰ is not an ESS.

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This literature was motivated mainly by the question which types of investment strategies will survive in the long run in a stochastic environment. Our focus is different: the structure of equilibrium in the static game.

The salient point is the coincidence of Nash and Walras independently of the number of players. This is quite different from the well-known observation (at least since Cournot) that Nash equilibria tend to Walrasian allocations in the limit, when the number of players goes to infinity, cf. [4] and the references given there.

For example, it would not be appropriate for the model of [5]. This model has constant supply of all assets, but features explicitly both strategic and non-strategic (competitive) traders. The non-strategic traders optimize portfolios consisting of the same assets as the strategic traders, the only difference being that they take prices as given. Clearly, the non-strategic traders' total demand for an asset depends on all prices in an essential way, and so does the remaining “supply” left for the strategic traders.

This interpretation was suggested by a referee.

This can be made precise, see [10].

A connection between evolutionary stability and competitive outcomes has also been found in other contexts ([9]).

For a precise statement see S.4 in the appendix.

Note on normalization: in the literature with fixed supply, it is frequently assumed (w.l.o.g.) that the total supply of each asset is equal to unity: S̄_k = 1 ∀k. In our context, a possible normalization is to choose units such that the return per unit of each asset is equal to unity: E_k = 1 ∀k. This is sometimes useful, e.g., in the context of Theorem 4.2.

The game G (or Ḡ) is not symmetric, see Section 5.

η′ = 0 is implied by η = 0 because η ≥ 0 and nondecreasing.

This argument parallels the one in [1]. Intuitively, as also pointed out in this paper, if an investor shifts money from one market to another, he “works against himself” by increasing the price of the asset which he is buying, and decreasing the price of the asset which he is selling.