Long-run heterogeneity in an exchange economy with fixed-mix traders

Abstract

We consider an exchange economy where agents have heterogeneous beliefs and assets are long-lived, and investigate the coupled dynamics of asset prices and agents’ wealth. We assume that agents hold fixed-mix portfolios and invest on each asset proportionally to its expected dividends. We prove the existence and uniqueness of a sequence of arbitrage-free market equilibrium prices and provide sufficient conditions for an agent, or a group of agents, to survive or dominate. Our main finding is that long-run coexistence of agents with heterogeneous beliefs, leading to asset prices endogenous fluctuations, is a generic outcome of the market selection process.

Proofs of Theorems and Lemmas

1.1 Proof of Lemma 1

Let $\bar{\alpha }=\max _{i\in N, k\in K} \{\alpha ^i_k\}$ and $\bar{\delta }=\max _{i \in N} \{ \sum _{k=1}^K \alpha ^i_k\}$. Consider the matrix $A(W;\alpha )$ in (9) whose element $A_{k,l}$ reads

$$\begin{aligned} A_{k,l}=\frac{\sum \nolimits _{i=1}^N\alpha ^i_k\alpha ^i_l W^i}{\sum \nolimits _{j=1}^N \alpha ^j_l W^j}\,. \end{aligned}$$

From $\mathbf{R1}$ and $\mathbf{R2}$, it follows that $0<A_{k,l}<\bar{\alpha }<1$ for all k, l, and that $0< \sum _{k=1}^K \, A_{k,l}< \bar{\delta } < 1$ for all l. The first group of inequalities implies

$$\begin{aligned} \left| \mathbb {I}-A \right| _{l,l}- \sum _{k=1,k\ne l}^K \, \left| \mathbb {I}-A \right| _{k,l}= \sum _{k=1}^K \, \left( \mathbb {I}-A \right) _{k,l}\,, \end{aligned}$$

while the second group implies $\sum _{k=1}^K \, \left( \mathbb {I}-A \right) _{k,l}=1-\sum _{k=1}^K \, A_{k,l}>1-\bar{\delta }>0$. Thus, the matrix $\mathbb {I}-A$ is column strictly diagonally dominant and, by the Levy–Desplanques theorem (Taussky 1949), invertible.

1.2 Proof of Proposition 1

The first part of the statement follows from Lemma 1 and from the derivation in the text before the proposition.

Regarding the absence of arbitrages consider the following. According to Stiemke’s Lemma, the absence of arbitrage is equivalent to the existence of a vector $q \in \mathbb {R}^S_{++}$ such that $R(W; \alpha , D) q = P$ or, with (12),

$$\begin{aligned} D q =(\mathbb {I}- A(W; \alpha ))P= \left[ \begin{matrix} 1-\frac{\sum \limits _{i=1}^N\alpha ^i_1\alpha ^i_1 W^i}{\sum \limits _{j=1}^N \alpha ^j_1 W^j} &{}\cdots &{} -\frac{\sum \limits _{i=1}^N\alpha ^i_1\alpha ^i_K W^i}{\sum \limits _{j=1}^N \alpha ^j_K W^j}\\ \vdots &{} \ddots &{}\vdots \\ -\frac{\sum \limits _{i=1}^N\alpha ^i_K\alpha ^i_1 W^i}{\sum \limits _{j=1}^N \alpha ^j_1 W^j} &{} \cdots &{} 1-\frac{\sum \limits _{i=1}^N\alpha ^i_K\alpha ^i_K W^i}{\sum \limits _{j=1}^N \alpha ^j_K W^j} \end{matrix} \right] \left[ \begin{matrix} \sum \limits _{j=1}^N \alpha ^j_1 W^j\\ \vdots \\ \vdots \\ \sum \limits _{j=1}^N \alpha ^j_K W^j \end{matrix} \right] . \end{aligned}$$

Thus, the kth component of $(\mathbb {I} - A(W; \alpha ))P$ reads $\sum _{i=1}^{N}[(1-\delta ^i) \delta ^i W^i]x_k^i$. By $\mathbf{R1}$ it is $x^i_k=\sum _{s=1}^S\pi ^i_s d_{k}(s)$; hence, one has

$$\begin{aligned} D q =D \left[ \begin{matrix} \sum \nolimits _{i=1}^{N}[(1-\delta ^i) \delta ^i W^i]\pi _1^i\\ \vdots \\ \sum \nolimits _{i=1}^{N}[(1-\delta ^i) \delta ^i W^i]\pi _S^i \end{matrix} \right] . \end{aligned}$$

That is, calling $q_s$ the sth component of q and given that D has full rank by $\mathbf{D4}$, it is $q_s=\sum _{i=1}^{N}[(1-\delta ^i) \delta ^i W^i]\pi ^i_s$. $\mathbf{R1}$, $\mathbf{R2}$, and the first part of the Proposition imply $q_s>0$ $\forall s\in S$ so that the statement is proven.

1.3 Proof of Propositions 2

As we shall show, the stochastic process that corresponds to the two groups’ relative wealth dynamics has bounded increments. As a result, we can prove the proposition by applying Theorem 2.1 in Bottazzi and Dindo (2015).

Consider the variable

$$\begin{aligned} z^I_t=\log \frac{w^I_t}{1-w^I_t} \end{aligned}$$

(25)

such that $z^I_t=z^I_{t-1}+ g^I(\sigma _t)$, with $g^I(\sigma _t)=\log G^I(\sigma _t)$ and

$$\begin{aligned} G^I(\sigma _t)= \frac{\sum _{k=1}^{K} r_{k,s_t}(w_{t-1};x,\delta ,D) \beta ^I_{k}(w_{t-1};x)}{\sum _{k=1}^{K} r_{k,s_t}(w_{t-1};x,\delta ,D) \beta ^{I^c}_{k}(w_{t-1};x)} \;. \end{aligned}$$

(26)

One has the following.

Lemma 5

The process $z^I_t$ has bounded increments, that is, there exists a $B \in \mathbb {R}$ such that $|z^I_t-z^I_{t-1}|<B$ ${{\mathrm{P}}}$-a.s..

Proof

By $\mathbf{R3}$ there exists a small enough $\varepsilon >0$ such that $\varepsilon \le x^i_{k}\le 1-\varepsilon$ $\forall \,i,k$. Since

$$\begin{aligned} \beta ^I_k(w_t;x)=\frac{x^I_k(w_t;x)}{\sum _{i=1}^N x^i_k w^i_t}\,, \end{aligned}$$

for any asset k and any time t it is

$$\begin{aligned} \frac{\varepsilon }{1-\varepsilon } \le \beta ^I_k(w_t;x)\le \frac{1-\varepsilon }{\varepsilon }\,. \end{aligned}$$

Consider the matrix

$$\begin{aligned} r_{k,s}(w_{t};x,\delta ,D)= & {} [(1-\delta )(\mathbb {I}-\delta A(w_{t};x))^{-1} \, D]_{k,s}\\= & {} (1-\delta )d_{k,s}+\delta p_{k,s}(w_{t};x,\delta ,D)\,. \end{aligned}$$

It holds: $0<r_{k,s}(w_{t};x,\delta ,D)<1$ $\forall k,t,s$ and $\sum _{k=1}^K r_{k,s}(w_{t};x,\delta ,D)=1$ $\forall s,t$. Thus for any group I, state s and time t one has

$$\begin{aligned} \frac{\varepsilon }{1-\varepsilon } \le \sum \limits _{k=1}^{K}r_{k,s}(w_{t};x,\delta ,D)\beta ^I_k(w_t;x) \le \frac{1-\varepsilon }{\varepsilon }\;. \end{aligned}$$

Given (25) and (26), by direct algebraic substitution it is straightforward to verify that

$$\begin{aligned} 2\log \frac{\varepsilon }{1-\varepsilon }\le z^I_{t}-z^I_{t-1} \le 2\log \frac{1-\varepsilon }{\varepsilon } \end{aligned}$$

and the statement is proven. $\square$

In order to prove Proposition 2, use the notation of Bottazzi and Dindo (2015) and call $\mu _t(z^I)$ the date t conditional drift of the process $\{z^I_t\}$, $\mu _t(z^I)={{\mathrm{E}}}^{{{\mathrm{P}}}}[z^I_{t+1}-z^I_t | \mathfrak {I}_t, z^I_t=z^I]$. If $\mu ^I(w;v)$ is an element of the set $\{ \mu ^I(w) | w\in \varDelta ^N,w^I=v\}$, with $\mu ^I(w)$ defined in (19), then by construction

$$\begin{aligned} \mu _t(z^I)=\mu ^I\left( w_t;\,v=\frac{e^{z^I}}{1+e^{z^I}}\right) . \end{aligned}$$

Having fixed $z^I$, and thus v, the precise value of $w_t$ still depends on the specific element of $\mathfrak {I}_t$ considered.

We shall start from the proof of statement (i). If $\underline{\mu }^I(0) > 0$, then, given the property of the lower bound and the continuity of $\mu ^I(w;v)$ with respect to v, there exist $\varepsilon >0$ and $M>0$ such that, for all $z^I<-M$ and t, it is $\mu _t(z^I)>\varepsilon$. Since $z^I_t$ has bounded increments, Theorem 2.1 in Bottazzi and Dindo (2015) applies and ${{\mathrm{Prob}}}{\{\limsup _{t\rightarrow \infty } z^I_t > -\infty \}}=1$. Provided $\overline{\mu }^I(1) > 0$, the same reasoning applies to prove (ii), see also Corollary 2.1 of Bottazzi and Dindo (2015).

1.4 Proof of Lemma 2

Let us consider the process $z_t^I$ in (25) and all the other quantities defined at the beginning of appendix A.3. We begin with the following Lemma.

Lemma 6

If the set of rules are not overlapping, $\mathbf{R4}$, and if there are no redundant assets, $\mathbf{D4}$, then $z_t^I$ does not possess any deterministic fixed point, that is, $\not \exists \;z$ s.t. $P (z^I_t=z|z^I_{t'}=z)=1$ $\forall t>t'$.

Proof

Suppose such z exists and at a certain time $t-1$ it is $z^I_{t-1}=z$. Then, by definition, it holds that $z^I_{t}-z^I_{t-1}=0$ for all the possible states of the world $s=1,2,...,S$, so that

$$\begin{aligned} \sum \limits _{k=1}^{K} r_{k,s}(w_{t-1};x,\delta ,D)\left( \beta ^I_{k}(w_{t-1};x)-\beta ^{I^c}_{k}(w_{t-1};x)\right) =0 \quad \forall s=1,2,...,S\;. \end{aligned}$$

That is

$$\begin{aligned} \left( \beta ^{I}(w_{t-1};x)-\beta ^ {I^c}(w_{t-1};x)\right) \, \left( (\mathbb {I}-\delta A(w_{t-1};x))^{-1} D\right) = \varvec{0} \;. \end{aligned}$$

The trivial solution $\beta ^I=\beta ^{I^c}$ is excluded by $\mathbf{R4}$, and according to Proposition 1, the kernel of $(\mathbb {I}-\delta A(w_{t-1},x))^{-1}D$ is zero, implying that the system of equations has no solution and the statement is proven. $\square$

The proof proceeds by noticing that $G^I$ in (26) depends on history $\sigma _t$ through the wealth distribution $w_t$ and the last realized state $s_t$. Given the distribution $w \in \varDelta ^{N}$ define

$$\begin{aligned} \bar{G}^I(\sigma _t)=\max _{s=1,\ldots ,S} \{|G^I(w,s_t)|\}\;, \end{aligned}$$

which, being the upper envelope of continuous functions, is a continuous function on the compact set $\varDelta ^{N}$. Then, by the Weierstrass theorem, it has a minimum $\underline{G}$. Moreover, it is $\underline{G}>0$ because, otherwise, $z^I_t$ would have a deterministic fixed point, which is not possible by Lemma 6. Then

$$\begin{aligned} {{\mathrm{Prob}}}\left\{ |z^i_{t}-z^i_{t-1}|\ge \underline{g} \,|\,\mathfrak {I}_{t-1}\right\} \ge \rho =\min \{\pi _1,\ldots ,\pi _S\}. \end{aligned}$$

and by taking $\gamma = {\min \lbrace \underline{g},\rho \rbrace }/{2}$ the assertion is proven.

1.5 Proof of Lemma 3

Let us consider the first statement. If it is wrong, then

$$\begin{aligned} \frac{w^{I^c}_{t+1}}{w^{I^c}_{t}}-\frac{w^{I}_{t+1}}{w^{I}_{t}}= \sum \limits _{k=1}^{K} \left( \beta ^{I^c}_k(w_{t};x)-\beta ^{I}_k(w_{t};x)\right) \, r_{k,s}(w_{t};x,\delta ,D)\ge 0 \quad \forall s \end{aligned}$$

(27)

and, since the process does not admit any deterministic fixed point (c.f. Lemma 6), the inequality is strict for some $s^\prime$. By construction it is

$$\begin{aligned} \sum \limits _{k=1}^{K} \left( \beta ^{I^c}_k(w_{t};x)-\beta ^I_k(w_{t};x)\right) p_{k,t} = 0\,. \end{aligned}$$

Together with (27) the latter implies that $\beta ^{I^c}(w_{t};x)-\beta ^I(w_{t};x)$ would be a weak arbitrage, which contradicts the hypotheses. For the second statement one can reason following the same lines and, in order to complete the proof, it is enough to choose $\epsilon =\min _s\{\pi _s\}/2$.

1.6 Proof of Proposition 3

Consider the process $z^I_t$ in (25) and all the other quantities defined at the beginning of appendix A.3. Then we can directly apply Lemma 5 and $z^I_t$ has bounded increments. $\mathbf{R4}$ and the lack of arbitrages imply by Lemma 3

$$\begin{aligned} {{\mathrm{Prob}}}{\left\{ z^I_{t+1}-z^I_t> 0\, \Big | \,\mathfrak {I}_t\right\} }= & {} {{\mathrm{Prob}}}{\left\{ \frac{w^I_{t+1}}{w^{I^c}_{t+1}}> \frac{w^{I}_{t}}{w^{I^c}_{t}}\, \Big | \,\mathfrak {I}_t\right\} }\\= & {} {{\mathrm{Prob}}}{\left\{ \frac{w^I_{t+1}}{w^{I}_{t}}> \frac{w^{I^c}_{t+1}}{w^{I^c}_{t}}\, \Big | \,\mathfrak {I}_t\right\} }>\epsilon \,. \end{aligned}$$

Moreover, invoking Lemma 2, one has

$$\begin{aligned} {{\mathrm{Prob}}}\left\{ \left| z^I_{t+1}-z^I_t\right|>\gamma \Big | \mathfrak {I}_{t}\right\} ={{\mathrm{Prob}}}\left\{ \left| \log \frac{w^I_{t+1}}{w^{I^c}_{t+1}}-\log \frac{w^{I}_{t}}{w^{I^c}_{t}}\right|>\gamma \, \Bigg | \mathfrak {I}_{t}\right\} > \gamma \,. \end{aligned}$$

Thus, defining $\epsilon _L=\min \{\epsilon ,\gamma \}/2$, one gets ${{\mathrm{Prob}}}\{z^I_{t+1}>z^I_t+\epsilon _L\,| \mathfrak {I}_{t}\}>\epsilon _L$, meaning that the process $z^I_t$ has finite positive increments. A symmetric argument shows that $z^I_t$ has also negative finite increments.

We shall start from the proof of statement (i) and (ii). Using the notation of Bottazzi and Dindo (2015), we call $\mu _t(z^I)$ the date t conditional drift of $\{z^I_t\}$, $\mu _t(z^I)={{\mathrm{E}}}^{{{\mathrm{P}}}}[z^I_{t+1}-z^I_t | \mathfrak {I}_t, z^I_t=z^I]$. If $\mu ^I(w;v)$ is an element of the set $\{ \mu ^I(w) | w\in \varDelta ^N,w^I=v\}$, with $\mu ^I(w)$ defined in (19), then by construction

$$\begin{aligned} \mu _t(z^I)=\mu ^I\left( w_t;\,v=\frac{e^{z^I}}{1+e^{z^I}}\right) \,. \end{aligned}$$

Having fixed $z^I$, and thus v, the precise value of $w_t$ still depends on the specific element of $\mathfrak {I}_t$ considered.

By continuity of $\mu ^I(w;v)$ with respect to v and by the definition of lower bound, the conditions $\underline{\mu }^I(0) > 0$ and $\underline{\mu }^I(1) > 0$ imply that there exist $\varepsilon >0$ and $M>0$ such that, for any t, it is $\mu _t(z^I)>\varepsilon$ if $z^I>M$ or $z^I<-M$. Using Theorem 3.1 from Bottazzi and Dindo (2015), it follows that ${{\mathrm{Prob}}}{\{\lim _{t\rightarrow \infty } z^I_t = +\infty \}}=1$, group I dominates. Conversely, conditions $\overline{\mu }^I(0) < 0$ and $\overline{\mu }^I(1) <0$ imply that there exist $\varepsilon >0$ and $M>0$ such that, for any t, it is $\mu _t(z^I)<-\varepsilon$ if $z^I>M$ or $z^I<-M$. Using Corollary 3.1 of Bottazzi and Dindo (2015), it follows that ${{\mathrm{Prob}}}{\{\lim _{t\rightarrow \infty } z^I_t = -\infty \}}=1$ and group I vanishes.

In order to prove statement (iii) note that, by Proposition 2, we already know that both groups survive. Thus, we need to prove that for $G=I, I^c$

$$\begin{aligned} {{\mathrm{Prob}}}\{\liminf _{t \rightarrow \infty } w^G_{t} = 0\,\, \text {and} \,\,\limsup _{t \rightarrow \infty } w^G_{t} = 1\}=1\,. \end{aligned}$$

Assume by contradiction that for a group G

$$\begin{aligned} {{\mathrm{Prob}}}\{\liminf _{t \rightarrow \infty } w^G_{t} = 0\,\, \text {and} \,\,\limsup _{t \rightarrow \infty } w^G_{t} = 1\}<1\,. \end{aligned}$$

Then, there must exist a positive measure set $\varSigma ' \subset \varSigma$ such that for all $\sigma \in \varSigma '$

$$\begin{aligned} \liminf _{t \rightarrow \infty } w^G_{t}(\sigma )> 0\,\, \text {or} \,\,\limsup _{t \rightarrow \infty } w^G_{t}(\sigma ) < 1\,. \end{aligned}$$

The latter is in contradiction with the process $\{z^I_t\}$ having finite positive and negative increments.

1.7 Proof of Corollary 2

Consider the group formed by agent i alone, where $\pi ^i=\pi$, and call $\mu ^i(w;v)$ a generic element of the set $\{ \mu ^i(w) | w\in \varDelta ^N,w^i=v\}$. When $v=1$ agent i owns all the wealth and normalized prices correspond to her portfolio rule. Thus, from (19) one gets

$$\begin{aligned} \mu ^i(w;1)=-\sum \limits _{s=1}^S\pi _s\log \left( \delta +(1-\delta )\sum \limits _{k=1}^K d_{k,s}\frac{x^{i^c}_k(w,x)}{x^i_k}\right) , \end{aligned}$$

where $x^{i^c}(w,x)$ is the aggregate portfolio rule of all the agents different from i. From the strict convexity of the function $-\log (\cdot )$ and by the fact that agent i has correct beliefs, one has

$$\begin{aligned} \mu ^i(w;1)>-\log \left( \delta +(1-\delta )\sum \limits _{k=1}^K \left( \sum \limits _{s=1}^S\pi _sd_{k,s}\right) \frac{x_k^{i^c}(w,x)}{x_k^i}\right) =0, \end{aligned}$$

where in the latter we have used the definition of Subjective Generalized Kelly rule, $\mathbf{R1}$. This immediately implies $\underline{\mu }^i(1)>0$.

Now consider the case $v=0$, one has

$$\begin{aligned} \mu ^i(w;0)=\sum \limits _{s=1}^S\pi _s\log \left( \sum \limits _{k=1}^K\frac{r_{k,s}(w;x,\delta ,D)}{x_k^{i^c}(w,x)} x^i_k\right) , \end{aligned}$$

where $r_{k,s}(w;x,\delta ,D)$ is the k, s element of the normalized payoff matrix $r(w;x,\delta ,D)$. To prove that $\underline{\mu }^i(0)>0$, we show that $\mu ^i(w;0)>0$ for any admissible w. First, we prove that $\lim _{\delta \rightarrow 0}\mu ^i(w;0)>0$, and then we show that $\lim _{\delta \rightarrow 1}\mu ^i(w;0)=0$.

For the first step one has

$$\begin{aligned} \frac{\partial \mu ^i(w;0)}{\partial \delta }= & {} \sum \limits _{s=1}^S\pi _s\left( \sum \limits _{k=1}^K\frac{r_{k,s}(w;x,\delta ,D)}{x^{i^c}_k(w,x)}x^i_k\right) ^{-1}\\&\left( \sum \limits _{k=1}^K \frac{x^i_k}{x^{i^c}_k(w,x)} \frac{\partial r_{k,s}(w;x,\delta ,D)}{\partial \delta }\right) . \end{aligned}$$

Its sign depends upon the derivative of the payoff matrix with respect to $\delta$. Using matrix differentiation, it reads

$$\begin{aligned} \frac{\partial r(w;x,\delta ,D)}{\partial \delta }= & {} -(\mathbb {I}-\delta A(w;x))^{-1}D\\&-(1-\delta )(\mathbb {I}-\delta A(w;x))^{-1} A(w;x) (\mathbb {I}-\delta A(w;x))^{-1}D\,, \end{aligned}$$

and implies $\partial r_{k,s}(w;x,\delta ,D)/\partial \delta <0$ for any k, s.

For the second step, we use that $r_{k,s}(w;x,\delta ,D)=(1-\delta )\,d_{k,s}+\delta \, p_{k,s}(w;x,\delta ,D)$. Call p the matrix whose k, s element is $p_{k,s}(w;x,\delta ,D)$. Let us rewrite equation (10) in terms of normalized quantities, it reads

$$\begin{aligned} (\mathbb {I}-\delta A(w;x)) p=(1-\delta )A(w;x)D\,. \end{aligned}$$

Thus in the limit of $\delta \rightarrow 1$ every column of p, call it $p_s$, solves the system

$$\begin{aligned} (\mathbb {I}-A(w;x))p_s=\mathbf{0}\,, \end{aligned}$$

which can be written as $A(w;x)p_s=p_s$. Notice that, given w with $w^i=0$, for each s $p_s=x^{i^c}(w,x)$ is a possible solution of the system. Moreover, since $\sum \limits _{k=1}^K |A_{k,j}(w;x)|=1$ $\forall j$ implies that A(w, x) is a contraction, by Banach fixed point theorem such solution is unique. Thus,

$$\begin{aligned} \lim _{\delta \rightarrow 1}\mu ^i(w;0)=\sum \limits _{s=1}^S\pi _s\log \left( \sum \limits _{k=1}^K\frac{x_k^{i^c}(w,x)}{x_k^{i^c}(w,x)}x^i_k\right) =0\,. \end{aligned}$$

Concluding we have shown that $\underline{\mu }^i(1)>0$ and $\underline{\mu }^i(0)>0$, hence by point (i) of Proposition 3 agent i dominates and the Corollary is proven.

1.8 Proof of Lemma 4

From the definition of conditional drift

$$\begin{aligned} \mu (0)= & {} \sum \limits _{s=1}^{S}\pi _s\log \left( \delta +(1-\delta )\sum \limits _{k=1}^{K}d_{k,s}\frac{x^1_k}{x^2_k}\right) \quad \text {and} \quad \\ \mu (1)= & {} -\sum \limits _{s=1}^{S}\pi _s\log \left( \delta +(1-\delta )\sum \limits _{k=1}^{K}d_{k,s}\frac{x^2_k}{x^1_k}\right) . \end{aligned}$$

Considering that $0\le d_{k,s} \le 1$ for all s, k and that $\sum _{k=1}^{K}d_{k,s}=1$ for all s, we have

$$\begin{aligned} \mu (0)>(1-\delta )\sum \limits _{s=1}^{S}\pi _s\log \left( \sum \limits _{k=1}^{K}d_{k,s}\frac{x^1_k}{x^2_k}\right) \ge (1-\delta )\sum \limits _{k=1}^{K}x^*_k \log \left( \frac{x^1_k}{x^2_k}\right) \end{aligned}$$

and at the same time

$$\begin{aligned} \mu (1)<-(1-\delta )\sum \limits _{s=1}^{S}\pi _s\log \left( \sum \limits _{k=1}^{K}d_{k,s}\frac{x^2_k}{x^1_k}\right) \le (1-\delta ) \sum \limits _{k=1}^{K}x^*_k \log \left( \frac{x^1_k}{x^2_k}\right) . \end{aligned}$$

Putting together the two inequalities proves the assertion.

1.9 Proof of Proposition 4

The statement follows from the particular case of Proposition 3 when $N=2$ together with Lemma 4 and the definition of normalized prices.

1.10 Proof of Corollary 3

The proof of the survival of agent 1 follows from Lemma 4 and Proposition 4. When agent 1 beliefs are correct, so that $x^1=x^*$, she also dominates since, exploiting the strict convexity of $-\log (\cdot )$,

$$\begin{aligned} \mu (1)&=\sum _s \pi _s \left( -\log \left( \delta +(1-\delta )\sum _{k=1}^{K} d_{k,s} \frac{x^2_k}{x^*_k}\right) \right) >\\&\quad \,\,-\log \left( \delta +(1-\delta )\sum _{k=1}^{K}\frac{x^2_k}{x^*_k}\sum _s\pi _s d_{k,s}\right) \\&=-\log \left( \delta +(1-\delta )\sum _{k=1}^{K}\frac{x^2_k}{x^*_k}x^*_k\right) =0\,. \end{aligned}$$

1.11 Proof of Proposition 5

The statement follows from the properties of the function $KL(x||x^*)\,:\,\varDelta ^K\rightarrow \mathbb {R}_+,\,x\mapsto KL(x||x^*)$. In particular, it is a continuous strictly convex function with a minimum equal to zero in $x=x^*$. Thus, it is defined over the compact set $\partial (\bar{\varDelta })$ and there exists a minimum over this set because of the Weierstrass theorem. The strict convexity of $KL(x||x^*)$ implies that it is also strictly quasi convex. This property together with the fact that $x^*\in \bar{\varDelta }$ implies $\{x\,:\,KL(x||x^*)<\mathcal {K}\}\subseteq \bar{\varDelta }$. Hence, it is possible to choose a $\pi ^1\ne \pi$ such that $KL(x^1||x^*)=m<\mathcal {K}-\epsilon$ with $\epsilon >0$ and small enough. Then, one can easily define the set $\varPi =\{\pi '\,:\,\pi '\in \varDelta ^S_{+},\, KL(x'||x^*)=m\}$ which has always at least two elements. Choosing $x^1$ and $x^2$ such that $\pi ^1,\pi ^2\in \varPi$ it is $\nabla _{x^*}(x^2||x^1)=0$.

1.12 Proof of Proposition 6

An asset market economy with agents trading according to Subjective Generalized Kelly rules and an asset market economy with agents maximizing expected log-utilities under effective beliefs have, by construction, the same relative wealth dynamics. When markets are dynamically complete and agents maximize an expected log-utility, there is no loss of generality, assuming that they are trading all possible contingent commodities in date zero. In fact, all asset structures, as long as they are complete, allow agents to achieve the same consumption allocation, so that the relative wealth dynamics does not depend upon the asset structure. Under time-zero trading, it is well known that agents allocate in each state contingent good a fraction of wealth proportional to the state likelihood. In a two-agent economy, the relative wealth dynamics can thus be rewritten as

$$\begin{aligned} \frac{w^1_{t+1}(\sigma _{t},s_{t+1})}{w^2_{t+1}(\sigma _{t},s_{t+1})} = \frac{\hat{\pi }^1_{s_{t+1}}(w_t;\delta ,D)}{\hat{\pi }^2_{s_{t+1}}(w_t;\delta ,D)} \frac{w^1_t(\sigma _t)}{w^2_t(\sigma _t)} \quad \forall \sigma _t, s_{t+1}, t\,. \end{aligned}$$

Applying the definition of the conditional drift $\mu (\cdot )$ to the log of the above process leads to the result.