Dynamic communication mechanism design

Abstract

We consider dynamic communication mechanisms in a quasi-linear environment with single-dimensional types. The mechanism designer gradually identifies agents’ valuations by iteratively offering prices to agents at different stages. Agents pay the maximum price they accepted if their desirable decision is made. We show that within weakly tight mechanisms, if a communication mechanism is ex-post incentive compatible, then it is a monotone-price mechanism. English auctions are characterized as a class of mechanisms that satisfy ex-post incentive compatibility and efficiency.

Appendices

Definition of extensive form games

In this appendix, we formally define DCM as extensive form games with perfect recall. A DCM is a tuple $\Gamma =( {\mathcal {H}}, \prec , J, p, A, (Q_i)_{i \in I}, g, m)$, where

• 1. ${\mathcal {H}}$ is a set of all histories or nodes, and $\prec$ is a partial order on ${\mathcal {H}}$ that represents precedence.

  • (a) The initial node is denoted by $h^0 \in {\mathcal {H}}$.

  • (b) The set of terminal histories is denoted by $H \equiv \{ h \in {\mathcal {H}} | \not \exists h', h \prec h' \}$.

  • (c) We suppose that $({\mathcal {H}}, \prec )$ is a binary tree. For every non-terminal history $h \in {\mathcal {H}} {\setminus } H$, there are two immediate successors.

  • (d) Let $H^t \subset {\mathcal {H}}$ be the set of histories with depth t. That is,

    $$\begin{aligned} H^t \equiv \{ h \in {\mathcal {H}} \mid |\{ h' \in {\mathcal {H}} | h' \prec h \} | =t \}. \end{aligned}$$

  • (e) $({\mathcal {H}}, \prec )$ has finite depth. Hence, there exists a number $K \in {\mathbb {N}}$ and

    $$\begin{aligned} {\mathcal {H}} = \bigcup _{t=0}^{K} H^t . \end{aligned}$$

  • (f) In the main text, a non-terminal history is often denoted by $h^t$ (using superscript), while a terminal history is denoted by h.

• 2. $J: {\mathcal {H}} {\setminus } H \rightarrow I$ is a player function, which assigns a player at each non-terminal history.

  • (a) For each $i \in I$, let ${\mathcal {H}}_i \equiv \{ h \in {\mathcal {H}} {\setminus } H \mid J(h) = i \}$ be the set of non-terminal histories that belong to i.

  • (b) In the main text, a mover at a non-terminal history $h^{t-1} \in H^{t-1}$ is often denoted by $J^t (h^{t-1})$.

• 3. $p: {\mathcal {H}} {\setminus } H \rightarrow {\mathbb {Z}}$ is a price function, which assigns a price offered to the mover $J (h^{t-1})$ at each non-terminal history.¹⁸

• 4. In each non-terminal history $h^t \in {\mathcal {H}} {\setminus } H$, the set of available actions at $h^t$ is given by $A = \{ yes, no \}$.

• 5. $Q_i$ is an information partition on ${\mathcal {H}}_i$.

  • (a) An information set is denoted by $q_i \subset {\mathcal {H}}_i$.

  • (b) We suppose that if $h, h' \in q_i$, then there exists some $t >0$ and $h,h' \in H^t$.

• 6. $g: H \rightarrow X$ is an allocation function, and $m: H \rightarrow {\mathbb {Z}}^n$ is a monetary transfer function.

Given a mechanism $\Gamma$, a pure strategy for agent i is a mapping $\sigma _i : V_i \rightarrow \{ yes, no \}^{{\mathcal {H}}_i}$ such that $\sigma _i (h) = \sigma _i (h')$ if h and $h'$ are in the same information set.

Proofs

1.1 Proof of Lemma 1

Suppose that a DCM is EPIC. The associated direct allocation rule is denoted by f. Suppose that there is an agent $i \in I$ and for some $v \in V$, $f(v) \in X_i$. The associated sincere history under v is denoted by h. Because of sincere reporting and the definition of the payment rule, the agent’s payment must be $m_i (h) \le v_i$.

Suppose that there exists ${\tilde{v}}_i >v_i$ and $f({\tilde{v}}_i, v_{-i}) \not \in X_i$. When agent i of type ${\tilde{v}}_i$ behaves as if his type is $v_i$, then the associated allocation is $f(v) \in X_i$ and the payment is $m_i (h)$. Hence, agent i’s deviating payoff is ${\tilde{v}}_i -m_i (h) >0$, which is a contradiction. $\square$

1.2 Proof of Lemma 2

Suppose that a DCM is EPIC. To have a contradiction, suppose that there exists a history h, and for some $i \in I(g(h))$ and some round t, $a_i^t =no$. Let t be the earliest such round, and consider $p_i^t$. For any state $v \in V(h)$, sincere reporting indicates $v_i <p_i^t$. Suppose ${\tilde{v}}_i \ge p_i^t$. By Lemma 1, $f({\tilde{v}}_i,v_{-i}) \in X_i$. By the construction of round t, the truthful history up to t is the same between v and $({\tilde{v}}_i,v_{-i})$.¹⁹ Hence, under the state $({\tilde{v}}_i,v_{-i})$, agent i is offered $p_i^t$ at round t and responds yes under sincere reporting. Thus, agent i pays at least $p_i^t$, and the payoff under sincere reporting is at most ${\tilde{v}}_i -p_i^t$. If agent i deviates and pretends to have $v_i$ under the state $({\tilde{v}}_i,v_{-i})$, the corresponding outcome is $f(v) \in X_i$ and the payment is strictly less than $p_i^t$, which contradicts EPIC. $\square$

1.3 Proof of Proposition 1

Suppose that there is a history such that agent i reports no at round s and is asked at a later round $t >s$. By the weak tightness, there exists a state $v \in V(h^{t-1})$ and $f(v) \in X_i$. However, this contradicts Lemma 2, which requires $f(v) \not \in X_i$. $\square$

1.4 Proof of Theorem 1

If part. Because an English auction is a monotone-price mechanism, it is EPIC. We will confirm that an English auction chooses the efficient outcome for every $v \in V$ under sincere reporting. To have a contradiction, suppose there exists a state $v \in V$ and an agent not having the highest valuation wins the auction. Let $I^* \subset I$ be the set of agents having the highest value under v. Let h be the associated terminal history. Because every agent $i \in I^*$ loses the auction, i is asked a price $p_i^t \ge v_i +1$ at some round t and responds no. Let agent i be the last agent who responds no among $I^*$ under h. By the properties of the auction rule, there exists an active agent $j \ne i$ and $p_j^{t-1} ={\bar{p}}_Y^{t-1} \ge p_i^t-1$ at round $t-1$. Because every agent in $I^*$ except for i is inactive after round $t-1$, agent j does not have the highest value. Hence, we have $v_j < v_i \le p_i^{t} -1 \le p_j^{t-1}$, which contradicts sincere reporting of agent j. Hence, the auction is efficient.

Only if part. Suppose that a DCM is EPIC and weakly tight. Then, it is a monotone-price mechanism. It is easy to see that the third property of the auction rule holds. When two or more agents remain at the termination, it is clear that the efficient outcome cannot be identified. So $|Y(h)| \le 1$ for all terminal history h. In addition, if $Y (h^t) = \{i \}$ and the designer asks agent i at round $t+1$, then the object is not allocated to anyone when i responds no. Hence, the mechanism terminates immediately if $|Y(h^t)|=1$.

Now, suppose that the first property does not hold for some $h^{t-1}$. Agent i is the unique active agent facing the highest current price of round $t-1$ and the mover at round t. Then,

$$\begin{aligned} p_i^t> p_i^{t-1} = {\bar{p}}_Y^{t-1} > p_j^{t-1} \end{aligned}$$

for all $j \in Y (h^{t-1}) {\setminus } \{ i \}$. Then a state v such that $v_i = p_i^{t-1}$, $v_j = p_j^{t-1}$ for each $j \in Y (h^{t-1}) {\setminus } \{ i \}$, and $v_k <p_k^{t-1}$ for each $k \in I {\setminus } Y(h^{t-1})$ is in the revealed state set $V(h^{t-1})$. The efficiency implies that agent i wins the auction, but he responds no at round t and loses, which is a contradiction. Hence, the first property holds.

Next, suppose that the second property does not hold for some $h^{t-1}$. Suppose that agent i is the mover at round t and $p_i^t \ge {\bar{p}}_Y^{t-1} +2$. Because $p_i^{t-1} \le {\bar{p}}_Y^{t-1}$, a state v such that $v_i = {\bar{p}}_Y^{t-1}+1$, $v_j = p_j^{t-1}$ for each $j \in Y (h^{t-1}) {\setminus } \{ i \}$, and $v_k <p_k^{t-1}$ for each $k \in I {\setminus } Y(h^{t-1})$ is in the revealed state set $V(h^{t-1})$. The efficiency implies that agent i wins the auction, but he responds no at round t and loses, which is a contradiction. Hence, the second property holds. $\square$

1.5 Proof of Theorem 2

By Lemma 2 and $I(x) \in \{ \emptyset , I \}$, the allocation $g(h) =1$ only if no agent reports no in the history h. By definition of DCM, a terminal history such that no agent reports no is uniquely determined and denoted by $h^*$. Let $J \subseteq I$ be the set of agents making a report in $h^*$ and let ${\bar{p}}_j$ be the maximum price offered to i in $h^*$. Then, the direct allocation rule is described as

$$\begin{aligned} f(v) ={\left\{ \begin{array}{ll} 1 &{} \text {if } (\forall j \in J), v_j \ge {\bar{p}}_j \\ 0 &{} \text {otherwise} \end{array}\right. }. \end{aligned}$$

This allocation rule is clearly obtained by a unanimous voting among J with offered prices $({\bar{p}}_j)_{j \in J}$. $\square$