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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 “multimarket games” in which several players compete in several arenas simultaneously, like multimarket Cournot oligopolies, or multiple rentseeking games.
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 rentseeking,
We study such a situation of “multimarket interaction” for the special case of
The present study was directly motivated by the model in [
It turns out that a slight modification of the model suffices to remove this anomaly: Rather than constant (inelastic) asset supply, we allow
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 restrictive
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 [
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,
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. Alternatively
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 constantsum.
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 nonsymmetric case. The results can be summarized as follows: In the
But the point of the paper is the
These somewhat counterintuitive deviations from the competitive outcome,
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 rate
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
We consider an asset market of the kind studied in [
It may be that
Each investor
The set
Alternatively (and equivalently), the behavior of an investor
For
even though, in the game to be considered below, investors are constrained to choose
Given
It is easy to see (
We conclude this section with a formula that will be useful in the sequel. Note that by definition (
Using
Expression
Given a strategy profile
If market
If
These data define the
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
This formulation of the game is quite natural. Indeed, investors frequently describe their strategy by stating how many
Our main interest will be in determining the structure of the
Given a strategy profile
A profile
In this case
There exists a unique competitive rate of return
For
Define
(i) There exists a unique profile
Let us call a profile
A profile
The asset market game
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
The results on the structure of equilibrium in the next section are based on the following observation. The marginal return to investor
This section contains the main results. Consider an equilibrium
Let
the largest investor (
the asset with the highest return (
if investor
i also holds all assets with higher or equal returns
all larger investors
larger investors hold relatively more lowyielding assets in the following sense: whenever
the lower the elasticity of supply for an asset, the higher its return:
larger investors have lower return rates:
Let
The proof of the various assertions in the Theorem is based on a careful examination of these firstorder conditions. Details are in the
Intuitively,
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
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 highestyielding 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 profitmaximizing to hold also the loweryielding assets, thus depressing his average return.
Let
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
Let
If the equilibrium satisfies
Assume that there exists a common elasticity function
The assumption of the Theorem means that all supply functions
We prove that all
A competitive equilibrium, if it exists, is symmetric. There may exist noncompetitive symmetric equilibria (
There is at most one Nash equilibrium in which every investor holds all assets,
Since in a symmetric equilibrium every investor must hold all assets, we obtain immediately:
There exists at most one symmetric Nash equilibrium.
Another interesting special case is when all investors have the same wealth,
Assume that all investors have the same wealth,
Consider an equilibrium and number investors and assets as in Theorem 4.1. By assumption, all investors have the same wealth, and by monotonicity (
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
Let
The concept of an evolutionarily stable strategy (ESS) for a finite game introduced by [
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 ([
Since the game
The game
For a strategy
The game
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,
The structure of equilibrium.
≥  ≥  …  ≥  ≥  ≥  


≤  ≤  …  ≤  ≤  …  ≤  row sums  

λ^{1} 


… 

… 


≤  ≤  ≥  ≥  ≥  
λ^{2}  0  … 

… 

… 


≤  ≤  ≥  ≥  ≥  
…  …  …  …  …  
λ 
0  0  … 

… 


≤  ≤  …  ≥  ≥  
…  …  …  …  …  
λ 
0  0  …  0  … 



 
column sums  …  … 
An equilibrium allocation: rows correspond to investors and columns to assets.
Nash equilibrium in Example 1.
 
λ^{1} = 0.25 



λ^{2} = 0.75 



 
I thank C. Alós  Ferrer, A. Ania, K. Podczeck and A. Ramsauer for helpful conversations.
Under assumption
for every
The function
define the function
For
Conversely, the properties of the price function
For future reference, we note that for any
(Since
The following is a more precise statement of the differentiability assumption in
For all
twice continuously differentiable on [0, ∞) with
twice continuously differentiable on (0, ∞), with
¿From now on, we maintain the assumptions
There are only two possible cases: either
supply is constant,
supply is not constant,
(α)
(β)
(γ) if
We omit the subscript
Clearly,
Assume now that supply is not constant. Then
Consider first the case of elastic supply at 0,
Both
Consider now the case of inelastic or unit elastic supply at 0, 0 <
Since
It implies also that
We want to show that
On the other hand, by (
Finally, since
If
The function
The price function
For
For
By Lemma A.2 (iii), the last term goes to zero for
This proves (i).
The price function
The game
The “if” part is trivial. Assume now that the game is constantsum,
Assume that all investors follow the proportional rule. Then
Summing this over
Since
To prepare for the proof of Theorem 3.1, note that the payoff functions
First we compute some derivatives. Let
Therefore, from (
By (
The formal proof of Theorem 3.1 is preceded by some lemmas.
For all
We have, on the convex set
Let
Moreover,
Fix an investor
That is, investor
As an immediate Corollary we have that all markets must be active at equilibrium.
If
By Lemma A.6,
We have to show that this situation is impossible at equilibrium.
Indeed, if (
On the other hand. there must exist an asset
Shifting a small amount
In particular, if an asset is in constant supply, then it must be held by more than one investor at equilibrium. Moreover, if
Assertions (ii) and (iii) follow from the two preceding Lemmas. It only remains to prove assertion (i) (existence of Nash equilibrium).
For
W.l.o.g. (passing to a subsequence if necessary) we may assume that the sequence
We claim that
Step 1.
Step 2. ∀
Assume, indirectly, that there is an asset
There must also exist some asset
By Step 1,
If the strategy profile (
Consider a Nash equilibrium
The following proof is based on a careful examination of the firstorder conditions for a Nash equilibrium. To understand the following arguments, it helps to keep
Let
Denote by
By Lemma A.7,
W.l.o.g., order the investors such that
We shall see below (see (
For given
It is easy to see that investor
The equilibrium allocation
The largest investor
let
Summing the first line in (
If
write
Now fix two investors
Since
Note that the coefficients
For
This implies
It remains to show that this implies (
Let
Defining
assume the contrary,
assume the contrary,
as before, assuming the contrary implies
Proceeding in this manner until
Let
Summing over
Write
Fix an investor
If
If
By Lemma 3.2 (ii) there exists a unique, strictly positive
Fix an investor
The crucial observation is that ∀
The deviator's payoff after the deviation is
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 nondeviator
This proves that
Let
First it is clear that if
Denote by
Denote by
The last term is positive for
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 wellknown observation (at least since Cournot) that Nash equilibria tend to Walrasian allocations in the limit, when the number of players goes to infinity,
For example, it would not be appropriate for the model of [
This interpretation was suggested by a referee.
This can be made precise, see [
A connection between evolutionary stability and competitive outcomes has also been found in other contexts ([
For a precise statement see
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:
The game
This argument parallels the one in [