Dynamic multilateral markets

Abstract

We study dynamic multilateral markets, in which players’ payoffs result from intra-coalitional bargaining. The latter is modeled as the ultimatum game with exogenous (time-invariant) recognition probabilities and unanimity acceptance rule. Players in agreeing coalitions leave the market and are replaced by their replicas, which keeps the pool of market participants constant over time. In this infinite game, we establish payoff uniqueness of stationary equilibria and the emergence of endogenous cooperation structures when traders experience some degree of (heterogeneous) bargaining frictions. When we focus on market games with different player types, we derive, under mild conditions, an explicit formula for each type’s equilibrium payoff as the market frictions vanish.

Appendix

For the sake of clarity we recall the shorthand notation $x(S)=\sum _{i\in S}x_i$ that will be used in all the analysis that follows.

Lemma 1

If $\delta _{k}=\delta$ for all $i\in \mathcal {N}$, then there is a threshold $\underline{\delta }<1$ such that for all $\delta \in ( \underline{\delta },1)$ the set of coalitions $\{S\subseteq \mathcal {N}: v(S)> \delta x^{\delta }(S)\}$ is the same and the $\mathbf {x}^{\delta }$ converges to a finite limit $\mathbf {x}^{1}$ as $\delta \rightarrow 1$.

Proof

In Theorem 1, we show that there is a unique solution $\mathbf {x}^{\delta }$ to (1) for any $\delta _{1}=\ldots =\delta _{N} =\delta \in (0,1)$. Let $\theta (S)$ be the agreement probability for the coalition $S$ in the LE $\mathbf {x}^{\delta }$, i.e., $\theta (S)=1$ if $v(S)\ge \delta x^{\delta }(S)$ and $\theta (S)=0$ if $v(S)< \delta x^{\delta }(S)$. We replace the max operators in (1) by the probabilities $\varvec{\theta }=\{\theta (S)\}_{S\subseteq \mathcal {N}}$. This converts (1) into a linear system with the unique solution $\mathbf {x} ^{\delta }$. This solution is given by the Cramer’s rule and equals the ratio of two determinants that are polynomials in $\delta$ of degree at most $N$,

$$\begin{aligned} x_{i}^{\delta }=P_{i}(\varvec{\theta },\delta )/Q(\varvec{\theta },\delta ). \end{aligned}$$

(15)

Note that $Q_{i}(\varvec{\theta },\delta )\ne 0$ for all $\delta \in (0,1)$ as the system (1) is nonsingular. By the same argument as in the proof of Proposition 1 in Manea (2011), it follows then that,

• (F1) The set $\overline{\Delta }$ of $\delta$ for which there is a coalition $S\subseteq \mathcal {N}$ such that $\delta x^{\delta }(S)=v(S)$ contains a finite number of elements. We define $\underline{\delta }=\max \overline{\Delta }$. Then, the function $v(S)-\delta x^{\delta }(S)$ is continuous in $\delta \in (\underline{\delta } ,1)$ as it is a rational function (ratio of two polynomials in (15)) and it has no roots outside $\overline{\Delta }$ by (F1). By Theorem 1, for any $\delta ^{0}>\underline{\delta }$ there is a unique solution $\mathbf {x}^{\delta ^{0}}$ to (1), which implies the agreement probabilities $\varvec{\theta }^{0}= \{\theta ^{0}(S)\}$. By the same argument as in the proof of Proposition 2 in Manea (2011) the following claims follow.

• (F2) For all $\delta >\underline{\delta }$, $v(S)>\delta x^{\delta }(S)$ if $\theta ^{0}(S)=1$ and $v(S)<\delta x^{\delta }(S)$ if $\theta ^{0}(S)=0$. The vector $\varvec{\theta }^{0}$ defines the limit cooperation structure.

• (F3) The vector of payoffs $\mathbf {x}^{\delta }$ converges to a finite limit $\mathbf {x}^{1}$ as $\delta \rightarrow 1$. $\square$

Lemma 2

In the PE $\mathbf {x}^{1}$,

$$\begin{aligned} \forall S,S^{\prime }\subseteq \mathcal {N}:\varvec{T}(S)\ne \varvec{T}(S^{\prime }),S\cap S^{\prime }\ne \varnothing ,\quad x^{1}(S)=v(S)\Rightarrow x^{1}(S^{\prime })>v(S^{\prime }). \end{aligned}$$

Proof

If ${x}^{1}(S)=v(S)$ and ${x}^{1}(S^{\prime })\le v(S^{\prime })$ then, by the definition (5), $\varvec{T}(S)=\varvec{T}(S^{\prime })$, which contradicts the statement $\varvec{T}(S)\ne \varvec{T}(S^{\prime })$. $\square$

Lemma 3

If the SE $\mathbf {x}^{\varvec{\delta }}$ (where $\varvec{\delta } =(\delta ,\ldots ,\delta )$) implies that all players of types $t$ and $s$ cooperate only in coalitions of type $\mathbf {n}=(n_{1},\ldots ,n_{T})$, then,

$$\begin{aligned} n_{s}\alpha _{s}{x}^{\delta }(\mathcal {N}_{t})=n_{t}\alpha _{t}{x}^{\delta }( \mathcal {N}_{s}). \end{aligned}$$

Proof

By summing up (1) over all $t$-type players for some $t\in \{1,\ldots ,T\}$, one obtains the total payoff to players of this type,

$$\begin{aligned} {x}^{\delta }(\mathcal {N}_{t})&=\delta {x}^{\delta }(\mathcal {N} _{t})+n_{t}\alpha _{t}\sum _{S:\varvec{T}(S)=\mathbf {n}}\pi _{S}\frac{v(S)-\delta {x}(S)}{\alpha (S)} \\&=\frac{n_{t}\alpha _{t}}{1-\delta }\sum _{S:\varvec{T}(S)=\mathbf {n}}\pi _{S} \frac{v(S)-\delta {x}(S)}{\alpha (S)}=:\frac{n_{t}\alpha _{t}}{1-\delta } \Delta ^{\delta }(\mathbf {n}). \end{aligned}$$

By the same argument, the total payoff to the players of type $s\in \{1,\ldots ,T\}$ is $x^{\delta }(\mathcal {N}_{s})= n_{s}\alpha _{s}\Delta ^{\delta }(\mathbf {n)}/(1-\delta )$ and the claim follows. $\square$

Proof of Theorem 2

• (i) First, we show that $v(S)\le {x}^{1}(S)$ for each coalition $S\subseteq \mathcal {N}$ in the LE $\mathbf {x^1}$. If there existed a coalition $S$ such that $v(S)>{x}^{1}(S)$, then $\pi _{S}>0$ due to our assumption in Sect. 4 and, by re-arranging (1) for a player $i\in S$ and taking the limit, we would obtain a contradiction,

  $$\begin{aligned} \underbrace{\lim _{\delta \rightarrow 1}(1-\delta )x_{i}^{\delta }}_{=0}= \underbrace{\lim _{\delta \rightarrow 1}\sum \nolimits _{S\in \mathcal {S} _{i}}\pi _{S}\dfrac{\alpha _{i}}{\alpha (S)}\max \{v(S)-\delta {x}^{\delta }(S),0\}}_{>0}, \end{aligned}$$

  (16)

  where $\delta =$ $\delta _{1}=\ldots$ $=\delta _{N}$. Furthermore, if $S$ is active in the LE $\mathbf {x^1}$, $v(S)>\delta {x} ^{\delta }(S)$ for all $\delta <1$ sufficiently close to one (Lemma 1) and, hence, (16) implies $v(S)={x}^{1}(S)$ for each active coalition $S$ in the LE $\mathbf {x^1}$. Now, in order to prove LE payoff equality for players of the same type, we assume, for the sake of contradiction, that $x_{i}^{1} >x_{j}^{1}$ for two distinct players of the same type, $\varvec{T}(\{i\})=\varvec{T}(\{j\})$, with $i\in S$ and $j\notin S$ for a coalition $S$ that is active in the LE $\mathbf {x^1}$. If $S^{\prime }:=\{j\}\cup S\setminus \{i\}$, then $\varvec{T}(S)=\varvec{T}(S^{\prime })$ and,

  $$\begin{aligned} {x}^{1}(S)=v(S)=v(S^{\prime })>{x}^{1}(S^{\prime })={x} ^{1}(S)-x_{i}^{1}+x_{j}^{1}, \end{aligned}$$

  which contradicts that $v(S^{\prime })\le {x}^{1}(S^{\prime })$ for any coalition $S^{\prime }\subseteq \mathcal {N}$ in the LE $\mathbf {x^1}$. Hence, we conclude that $x_{i}^{1}=x_{j}^{1}$ if $\varvec{T}(\{i\})=\varvec{T}(\{j\})$.

• (ii) We showed that $\mathbf {x}^{1}(S)\ge v(S)$ (with equality of active coalitions) and that all players of the same type receive equal limit payoffs in (i). From these results, it follows directly that $\sum _{t}n_{t}x_{t}^{1}= v(\mathbf {n})$ for each active coalition of type $\mathbf {n}=(n_{1},\ldots ,n_{T})$.

• (iii) Consider a player of type $t$ who cooperates in the LE $\mathbf {x^1}$ in a coalition $S$ with $\varvec{T}(S)=\mathbf {n}$. Then, by (i), we can write

  $$\begin{aligned} {x}^{1}(S)=v(S)=v(\mathbf {n})=\sum _{t=1}^{T}n_{t}x_{t}^{1},\quad {x} ^{1}(S^{\prime })\ge v(S^{\prime }),\quad \forall S^{\prime }\subseteq \mathcal {N}. \end{aligned}$$

  The latter inequality must hold, in particular, for

  $$ \begin{aligned} S^{\prime }=S^{-}:\varvec{T}(S^{-})=\mathbf {n}-\mathbf {e}_{t}\quad \& \quad S^{\prime }=S^{^{+}}:\varvec{T}(S^{+})=\mathbf {n}+\mathbf {e}_{t}. \end{aligned}$$

  whenever $v(\mathbf {n}-\mathbf {e}_{t})$ and $v(\mathbf {n}+\mathbf {e}_{t})$ are well-defined values. Then,

  $$\begin{aligned} {x}^{1}(S^{-})&={\textstyle \sum _{s\ne t}}n_{s}x_{s}+(n_{t}-1)x_{t}=v( \mathbf {n})-x_{t}\ge v(\mathbf {n}-\mathbf {e} _{t})=v(S^{-}), \\ {x}^{1}(S^{+})&={\sum _{s\ne t}}n_{s}x_{s}+(n_{t}+1)x_{t}=v(\mathbf {n} )+x_{t}\ge v(\mathbf {n}+\mathbf {e}_{t})=v(S^{+}), \end{aligned}$$

  which yields the claim. $\square$

Proof of Theorem 3

• (i) To show that in the PE, all players of the same type cooperate in MCs of homogeneous types, we will proceed by establishing a contradiction. Consider the PE $\mathbf {x^1}$ and assume, for the sake of contradiction, that in $\mathbf {x}^{1}$ players $i$ and $j$ of type $t$ cooperate in coalitions $S$ and $S^{\prime }$, respectively, with $\varvec{T}(S)=(n_{1},\ldots ,n_{T})\ne (n_{1}^{\prime },\ldots ,n_{T}^{\prime })=\varvec{T}(S^{\prime })$. Let the coalition $S^{\prime \prime }$ be the same as $S^{\prime }$ except for player $i$ who replaces player $j$, i.e. $S^{\prime \prime }=S^{\prime }\cup \{i\}\setminus \{j\}$. Hence, $\varvec{T}(S)\ne \varvec{T}(S^{\prime })=\varvec{T}(S^{\prime \prime })$ and $\{i\}\in S\cap S^{\prime \prime }$. As $S$ and $S^{\prime }$ are active, Theorem 2, items (i) and (ii), and the fact that $\varvec{T}(S^{\prime })=\varvec{T}(S^{\prime \prime })$ imply,

  $$\begin{aligned} v(S)={x}^{1}(S),\quad {x}^{1}(S^{\prime })=v(S^{\prime })=v(S^{\prime \prime })={x}^{1}(S^{\prime \prime }). \end{aligned}$$

  However, this contradicts, by Lemma 2, the partitioning property of $\mathbf {x^1}$ as $S\cap S^{\prime \prime }\ne \varnothing$ and $\varvec{T}(S)\ne \varvec{T}(S^{\prime \prime })$.

• (ii) Let $\delta =\delta _{1}=\ldots =\delta _{N}$ and consider the PE $\mathbf {x^1}$. We have established that each player of type $t$ cooperates in MCs of homogeneous types. By Lemma 3, the total payoff for all players of type $t$ and all players of type $s$, that cooperate in MCs of the same type as a type $t$-player, i.e., $\mathbf {n}=(n_{1},\ldots ,n_{T})$, satisfy

  $$\begin{aligned} n_{s}\alpha _{s}{x}^{\delta }(\mathcal {N}_{t})=n_{t}\alpha _{t}{x}^{\delta }( \mathcal {N}_{s}). \end{aligned}$$

  This equality must hold also for the PE $\mathbf {x}^{1}$. Thus using Theorem 2(i), we can re-write the last equality as

  $$\begin{aligned} n_{s}\alpha _{s}N_{t}x_{t}^{1}=n_{t}\alpha _{t}N_{s}x_{s}^{1}. \end{aligned}$$

  (17)

  In particular, $x_{t}^{1}=0$ implies $x_{s}^{1}=0$ for any two distinct types $s$ and $t$ that cooperate in a coalition $S:\varvec{T}(S)=\mathbf {n}$. This is only possible in PE if $v(S)=v(\mathbf {n})=0$. Suppose $x_{t}^{1}>0$ for some type $t$. Then, by Theorem 2 items (i) and (ii), it follows that

  $$\begin{aligned} v(S)=v(\mathbf {n})={x}^{1}(S)=\sum _{s=1}^{T}n_{s}x_{s}^{1}. \end{aligned}$$

  Using (17) to substitute for $x_{s}^{1}$,

  $$\begin{aligned} v(\mathbf {n})=\sum _{s=1}^{T}\frac{n_{s}^{2}\alpha _{s}N_{t}x_{t}^{1}}{ n_{t}\alpha _{t}N_{s}}=x_{t}^{1}\frac{N_{t}}{n_{t}\alpha _{t}}\sum _{s=1}^{T} \frac{n_{s}^{2}\alpha _{s}}{N_{s}}. \end{aligned}$$

  By re-arranging the above expression, we obtain the payoff of each type-$t$ player in the PE $\mathbf {x^1}$,

  $$\begin{aligned} x_{t}^{1}=v(\mathbf {n})\frac{n_{t}\alpha _{t}/N_{t}}{ \sum _{s=1}^{T}(n_{s}^{2}\alpha _{s}/N_{s})}. \end{aligned}$$

  $\square$

Proof of Theorem 4

$\mathbf {PE}\Rightarrow$ (7): In a PE, there is a partition $\mathcal {P}$ of player types and all types in the partition element $P\in \mathcal {P}$ cooperate in coalitions of a unique type $\mathbf {n}^{P}$ obtaining payoffs $\{x_{t}^{1}\}_{t\in P}$ computed by the formula (6). Then, by the LE condition 2(ii),

$$\begin{aligned} \sum _{P\in \mathcal {P}}\sum _{t\in P}m_{t}x_{t}^{1}&> v(\mathbf {m}),\quad \forall \mathbf {m\notin }\{\mathbf {n}^{P}\}_{P\in \mathcal {P}},\quad \mathbf {m\ne 0}. \\ \mathrm{{where}},\quad x_{t}^{1}&= \frac{v(\mathbf {n}^{P})n_{t}^{1}\alpha _{t}/N_{t}}{ \sum _{s\in P}(\alpha _{s}n_{s}^{2}/N_{s})},\quad t\in P. \nonumber \end{aligned}$$

(18)

We observe that the condition (18) is equivalent to (7).

(7)$\Rightarrow \mathbf {PE}$: The condition (7) can be written as,

$$\begin{aligned} \sum _{P\in \mathcal {P}}\sum _{t\in P}m_{t}x_{t}^{1}&> v(\mathbf {m}),\quad \forall \mathbf {m\notin }\{\mathbf {n}^{P}\}_{P\in \mathcal {P}},\quad \mathbf {m\ne 0}. \\ \mathrm{{where}},\quad x_{t}^{1}&= \frac{v(\mathbf {n}^{P})n_{t}^{1}\alpha _{t}/N_{t}}{ \sum _{s\in P}(\alpha _{s}n_{s}^{2}/N_{s})},\quad t\in P. \nonumber \end{aligned}$$

(19)

On the other hand,

$$\begin{aligned} \sum _{t=1}^{T}\mathbf {n}_{t}^{P}x_{t}^{1}=\sum _{t\in P}\frac{v(\mathbf {n} ^{P})n_{t}^{2}\alpha _{t}/N_{t}}{\sum _{s\in P}(\alpha _{s}n_{s}^{2}/N_{s})} =v(\mathbf {n}^{P}),\quad P\in \mathcal {P}. \end{aligned}$$

Hence, $\{\mathbf {n}^{P}\}_{P\in \mathcal {P}}$ are the only productive profiles that satisfy the LE condition 2(ii) with equality when the LE payoffs are computed by the formula (6). Therefore, $\{\mathbf {n}^{P}\}_{P\in \mathcal {P}}$ are the only active profiles. We note further that any coalition of type $\mathbf {n}^{P}$ has no intersection with a coalition of type $\mathbf {n}^{P\prime }$ if $P,P^{\prime }\in \mathcal {P}$ and $P\ne P^{\prime }$. Hence, any two active coalitions intersect only if they are of the same type. We conclude then that the LE is partitioning and the Theorem 3 confirms that the LE payoffs are given by (6).   $\square$