Efficient Mechanisms under Unawareness

Since mechanism design is closely related to contract theory, our paper contributes to the recent literature on contracting under unawareness (e.g., Lee (2008); Sommer and Loch (2009); von Thadden and Zhao (2012); Auster (2003); Filiz-Ozbay (2012); Grant et al. (2012); Chung and Fortnow (2016); Auster and Pavoni (2024); Lei and Zhao (2021); Francetich and Schipper (2024)). For instance, Francetich and Schipper (2024) show that screening contracts may not provide sufficient incentives to agents to reveal their awareness. Consequently, the principal may not be able to consider the full set of incentive constraints. In contrast, we show that we can go beyond incomplete contracts and reveal awareness in dynamic direct elaboration mechanisms that provide sufficient incentives for truthful reporting and raising awareness. However, we focus on efficient mechanisms rather than optimal mechanisms, affording us the opportunity of working in the belief-free paradigm of (conditional) dominant strategy implementation and allowing us to conveniently bypass the subtleties of updating beliefs under dynamic awareness. The literature on unawareness in contracting is very different from the earlier literature on indescribable contingencies in contracting. For instance, in Maskin and Tirole (1999), agents are fully aware of every particular payoff consequence but cannot ex-ante describe them. In contrast, under unawareness agents may not be fully aware of all payoff consequences in contracting. Closer to mechanism design, Li and Schipper (2024) study the seller’s decision to raise bidders’ awareness of characteristics before a second-price auction with entry fees. Optimal entry fees capture an additional unawareness rent due to unaware bidders misperceiving their probability of winning and the price to be paid upon winning. In contrast to our setting, the auctioneer is aware of everything ex-ante. Piermont (2024) studies how a decision maker can incentivize an expert to reveal novel aspects about a decision problem via an iterated revelation mechanism in which at each round the expert decides on whether or not to raise awareness of a contingency that in turn the decision maker considers when proposing a new contract that the expert can accept or reject. This bears some similarity with our dynamic direct elaboration mechanisms. Most importantly, we focus on a multi-agent setting with transferable utilities. Piermont (2024) shows that iterated revelation allows for efficient outcomes to emerge.

Herweg and Schmidt (2020) study a procurement problem with a principal and two agents who may be aware of some design flaws. Our paper differs from theirs in many respects: First, in Herweg and Schmidt (2020) agents can raise awareness of realized design flaws, agent’s private costs are independent of the design flaw, and fixing the design flaw requires a known common cost that is interim verified by an industry expert. In our model, agents report payoff types, can raise awareness of potential factors affecting payoffs, and individually elaborate how the payoffs change in light of new awareness. Second, Herweg and Schmidt (2020) construct an efficient direct mechanism under common awareness and then argue that there is an indirect mechanism that under asymmetric awareness that gives rise to the same outcomes and incentives. This leads them to conclude that there are also corresponding equilibria in these two mechanisms. Yet, an appropriate notion of equilibrium in mechanisms under asymmetric awareness should verify equilibrium behavior w.r.t what outcomes agents anticipate and how behavior is affect by changing anticipations of outcomes during the play. Our approach is more “direct”: We conduct our entire analysis of efficient conditional dominant strategy implementation in a direct mechanisms under asymmetric awareness thereby modeling every potential deviation from equilibrium behavior from the agent’s point view. Finally, Herweg and Schmidt (2020) focus on the particular but very relevant application to procurement while we consider more generally the efficient mechanism design problem under asymmetric awareness.

We study the design of efficient mechanisms in which raising awareness by one agent can via the mediator awareness and thus valuations of other agents. That is, even though we work with private valuations, valuations become endogenously interdependent. We know from Jehiel and Moldovanu (2001) that it is generically impossible to implement ex-post efficient outcome functions in settings with interdependent valuations. Why are we able to nevertheless implement efficiently? The key is that the interdependence in payoffs stems from awareness, which agents can only misreport in one direction. By using transfers that are sufficiently increasing in reported awareness, we incentivize agents to report as much awareness as possible — namely, their true awareness level. A related result is given by Krähmer and Strausz (2024). We discuss the connection further after the results.

We make use of unawareness type spaces introduced by Heifetz et al. (2006, 2013b) as well as games with unawareness developed by Heifetz et al. (2013a); Meier and Schipper (2014), and Schipper (2021). For alternative approaches, see Fagin and Halpern (1988); Halpern and Rêgo (2014); Feinberg (2021), and Board and Chung (2021).

Let $L$ be a finite lattice with order $\trianglerighteq$. Elements of the lattice represent awareness levels. The join of the lattice is denoted by $\bar{\ell}\in L$. Define $L(\ell):=\{\ell^{\prime}\in L:\ell^{\prime}\trianglelefteq\ell\}$ for $\ell\in L$. This is the sublattice of $L$ with join $\ell$. The significance of $L(\ell)$ is that an agent with awareness level $\ell$ can only reason about awareness levels in $L(\ell)$.

Fix a nonempty finite set of agents $I$. For each agent $i\in I$, there a collection of nonempty disjoint payoff type spaces $\{T_{i}^{\ell}\}_{\ell\in L}$. Let $\mathcal{T}_{i}:=\bigcup_{\ell\in L}T_{i}^{\ell}$. A payoff type $t_{i}\in\mathcal{T}_{i}$ is more than just a value for an object. If $t_{i}\in T_{i}^{\ell}$, then describing the payoff type also requires at least awareness level $\ell$. We illustrate this feature with the following example.

For any agent $i\in I$ and awareness levels $\ell,k\in L$ with $k\trianglerighteq\ell$, we require a surjective projection $r^{k}_{\ell}:T_{i}^{k}\longrightarrow T_{i}^{\ell}$ such that for all $\ell,\ell^{\prime},\ell^{\prime\prime}\in L$, $\ell^{\prime\prime}\trianglerighteq\ell^{\prime}\trianglerighteq\ell$, $r^{\ell^{\prime}}_{\ell}(r^{\ell^{\prime\prime}}_{\ell^{\prime}}(t_{i}))=r^{\ell^{\prime\prime}}_{\ell}(t_{i})$ and $r^{\ell}_{\ell}(t_{i})=t_{i}$. For brevity, we do not index $r^{k}_{\ell}$ by agents. The projection relates payoff types across awareness levels. Before we illustrate this notion, we also define extensions of payoff types to greater awareness levels. For any agent $i\in I$, awareness level $\ell\in L$, and payoff type $t_{i}\in T_{i}^{\ell}$, we let $t_{i}^{\uparrow}:=\bigcup_{k\trianglerighteq\ell,k\in L}(r_{\ell}^{k})^{-1}(t_{i})$. That is, $t_{i}^{\uparrow}$ is the union of inverse images at (weakly) greater awareness levels of payoff type $t_{i}$. Similarly, for any subset of payoff types in a given payoff type space, we let superscript “$\uparrow$” indicate the union of inverse images in payoff type spaces corresponding to greater awareness levels. We illustrate these notions in our prior example.

For any agent $i\in I$, we define $\lambda:\mathcal{T}_{i}\longrightarrow L$ by $\lambda(t_{i})=\ell$ if $t_{i}\in T_{i}^{\ell}$. Again, for brevity we do not index $\lambda$ by agents. The function $\lambda$ indicates the awareness level that is required to describe the payoff type. For instance, in Example 1, $\lambda(t_{1})=\{a,b\}$ while $\lambda(t_{1}^{\prime})=\{a,b,c\}$.

Our framework is general enough to capture payoff types described by all kinds of formal objects like sets of formulae in a formal language (e.g., Heifetz et al. (2008)), abstract sets, vectors, matrices, formal concepts, pre-sheaves etc. that may represent verbal, quantitative, or pictorial features of proposals, tenders, quotations, messages etc. in business practice. By abstracting from these particular features, we obtain a theory that works for more than one kind of formalism, ensure tractability, and focus on what is really essential for modeling awareness levels, namely the existence of an order of expressiveness of descriptions.

For each agent $i\in I$, payoff types and awareness levels are drawn consistently as follows: At awareness level $\bar{\ell}$, nature draws a payoff type $\bar{t}_{i}\in T_{i}^{\bar{\ell}}$ in the upmost payoff type space, interpreted as agent $i$’s true payoff type if she were aware of everything, and an awareness level $\ell_{i}\in L$. Consequently, the agent’s perceived payoff type is $r^{\bar{\ell}}_{\ell_{i}}(t_{i})$. That is, agent $i$ can “miss something” but he cannot perceive the “wrong” payoff type w.r.t. what he is aware.

More generally, for any awareness level $\ell\in L$, nature draws agent $i$’s corresponding type $r^{\bar{\ell}}_{\ell}(\bar{t}_{i})$ and agent $i$’s awareness level $\ell^{\prime}=\ell_{i}\wedge\ell$. This means in particular, that if $\ell=\ell_{i}$, then it is payoff type $r^{\bar{\ell}}_{\ell_{i}}(t_{i})$ and awareness level $\ell_{i}$ as just discussed. If $\bar{\ell}\trianglerighteq\ell\trianglerighteq\ell_{i}$, nature draws agent $i$’s corresponding type $r^{\bar{\ell}}_{\ell}(\bar{t}_{i})$ and awareness level $\ell_{i}$. If $\ell\in L$ with $\ell_{i}\trianglerighteq\ell$, nature draws agent $i$’s corresponding type $r^{\bar{\ell}}_{\ell}(\bar{t}_{i})$ and awareness level $\ell$. Finally, if $\ell\in L$ with $\ell\not\trianglerighteq\ell_{i}$, nature draws agent $i$’s corresponding type $r^{\bar{\ell}}_{\ell}(\bar{t}_{i})$ and awareness level $\ell^{\prime}=\ell_{i}\wedge\ell$. This specifies perceived payoff types and awareness levels consistently across the type spaces $\{T_{i}^{\ell}\}_{\ell\in L}$. Notice that if $t_{i}\in T_{i}^{\ell}$ for some $\ell\in L$, then the agent’s perceived type is always in a type space with (weakly) less awareness than $\ell$.

Figure 1 illustrates the payoff type structure. There are four awareness levels $L=\{\bar{\ell},\ell^{\prime},\ell^{\prime\prime},\underline{\ell}\}$ with $\bar{\ell}\triangleright\ell^{\prime}\triangleright\underline{\ell}$ and $\bar{\ell}\triangleright\ell^{\prime\prime}\triangleright\underline{\ell}$. Associated with each awareness level is a payoff type space for agent $i$. Projections from richer type spaces to poorer type spaces are indicated by dashed lines. Suppose that the payoff type selected by nature in the upmost payoff type space $T_{i}^{\bar{\ell}}$ is $\bar{t}_{i}^{1}$ indicated in red. Then the corresponding payoff types selected in the other payoff type spaces follow by projections and are also indicated in red. When the awareness level of agent $i$ selected by nature is $\ell_{i}=\ell^{\prime}$, then $t_{i}^{\prime}$ is the payoff type perceived by agent $i$. This is indicated with the red circle in type space $T_{i}^{\ell^{\prime}}$.

For any $\ell\in L$, let $\bm{T}^{\ell}:=\times_{i\in I}T_{i}^{\ell}$. Moreover, let $\boldsymbol{\mathcal{T}}:=\times_{i\in I}\mathcal{T}_{i}$. Similarly, we let $\bm{T}^{\ell}_{-i}:=\times_{j\in I\setminus\{i\}}T_{j}^{\ell}$ and $\boldsymbol{\mathcal{T}}_{-i}:=\times_{j\in\in I\setminus\{i\}}\mathcal{T}_{j}$.

When agents have the payoff type profile $\bm{t}=(t_{i})_{i\in I}\in\boldsymbol{\mathcal{T}}$, their pooled awareness level is $\widecheck{\lambda}(\bm{t}):=\bigvee_{i\in I}\lambda(t_{i})\in L$, i.e., the join of all agent’s awareness levels at payoff type profile $\bm{t}$. Since $L$ is a finite lattice, the join always exists in $L$. Note that $\widecheck{\lambda}(\bm{t})$ may be a greater awareness level than any of the agent’s awareness levels.

For each $\ell\in L$, there is a nonempty compact set of outcomes or allocations $X^{\ell}\subseteq X$. This formulation allows both for awareness of outcomes and awareness that some outcomes are infeasible.

Each agent $i\in I$ has an upper semi-continuous utility function $u_{i}:\bigcup_{\ell\in L}X^{\ell}\times T_{i}^{\ell}\longrightarrow\mathbb{R}$. From this formulation it is clear that we focus on private payoff types.

To facilitate a commonly used notion of efficiency, we assume that each agent’s utility function is quasilinear. I.e., for each $i\in I$, $u_{i}(x,t_{i}):=v_{i}(x_{0},t_{i})+x_{i}$ for $x=(x_{0},x_{1},...,x_{|I|})\in X^{\ell}:=X_{0}^{\ell}\times\mathbb{R}^{|I|}$ and $t_{i}\in T_{i}^{\ell}$, $\ell\in L$. As usual, $x_{0}$ describes the physical properties of the outcome while $(x_{i})_{i\in I}$ represents the vector of transfers made to agents.

We denote the outcome function by $f_{0}:\boldsymbol{\mathcal{T}}\longrightarrow X_{0}$. That is, $f_{0}(\bm{t})$ is the physical outcome prescribed by the outcome function $f_{0}$ to type profile $\bm{t}$. We require that for any $\bm{t}\in\boldsymbol{\mathcal{T}}$, $f_{0}(\bm{t})\in X_{0}^{\widecheck{\lambda}(\bm{t})}$. That is, the social planner can use joint awareness to select the outcome/allocation. If the reported payoff type profile is $\bm{t}$, then the planner’s awareness is $\widecheck{\lambda}(\bm{t})$ and hence outcomes in $X_{0}^{\widecheck{\lambda}(\bm{t})}$ are selected.⁴⁴4Typically the planer is not considered as an agent in mechanism design and hence we do not explicitly consider the awareness that the planner may have. Yet, all of our results remain intact when the planner joins her awareness together with the awareness of all agents whenever communicating the pooled awareness level in the dynamic elaboration mechanisms introduced in the next section. The formulation also makes clear that the social planner cannot necessarily describe the outcome function to agents in advance if the social planner is unaware of something herself. However, we assume that the social planner can commit to abstract properties of outcome functions such as efficiency.

We generalize utilitarian ex-post efficiency of outcome functions to unawareness by requiring it for every awareness level.

In standard mechanism design with quasilinear utilities and private values but without asymmetric unawareness, efficient implementation is almost equivalent to the use of Vickrey-Clark-Groves (VCG) mechanisms. We demonstrate that VCG mechanisms fail to implement efficiently under unawareness, quasilinear utilities, and private values.

There is a single object to be allocated. For simplicity, we assume that the value of not getting the object is always zero. The value of getting the object depends on the payoff type. It is written in blue above each payoff type, representing $v_{i}$. Consider now a payoff type profile $(t_{1}^{{}^{\prime\prime\prime\prime}},t_{2}^{{}^{\prime\prime\prime\prime}})$ and awareness levels $\ell_{1}=\bar{\ell}$ and $\ell_{2}=\underline{\ell}$ for agents 1 and 2, respectively. In Figure 2, we print the true payoff types in red. Yet, since agents differ in their awareness, we also indicate their corresponding perceived payoff types by red ovals. That is, agent 1 perceives his payoff type to be equal to his actual payoff type $t_{1}^{{}^{\prime\prime\prime\prime}}$, while agent 2 perceives his payoff type to be $t_{2}^{\prime}$ because his awareness level is $\underline{\ell}$.

Consider running a standard Vickrey second price auction in which the highest bidder wins the object and pays the second highest bid. In the dominant strategy equilibrium of the auction, agents bid truthfully their own perceived payoff type. That is, agent 1 bids 2 and agent 2 bids 1. Consequently, agent 1 obtains the object and pays a price of 1. His (net) utility is 1. Note that awareness is not raised (since agents just submit bids consisting of a value for the object). Clearly, the auction is not efficient since it does not allocate the object to the bidder with the highest true value for the object, which would be bidder 2.

If bidder 1 would have raised awareness of $\bar{\ell}$, then bidder 2 would realize that her value is 3. In such a case, bidder 2 might have bid 3, obtained the object, and paid a price of 2. This leaves bidder 1 with a utility 0, which is worse than his utility of 1 in the Vickrey auction. Thus, in the second price auction, bidder 1 has no incentive to raise bidder 2’s awareness.

In order to pool awareness among agents and allow agents to provide information on issues they are or became aware, we allow for communication not just from agents but also to agents. That is, we envision a mediator who receives messages about payoff types from agents, pools the awareness contained in those messages, and sends back messages to agents that potentially raise the awareness of agents to the pooled awareness level. Subsequently, agents may want to elaborate on their prior messages at least to the details of the pooled awareness level. This procedure may be repeated till no agent wants to elaborate further. To this end, we introduce a new class of dynamic mechanisms.

At the first stage, every agent reports a type. At later stages, agents can elaborate on their prior reported types. When agents report the payoff type profile $\bm{t}^{n-1}$ in stage $n-1$, the mediator awareness level is the pooled awareness level $\widecheck{\lambda}(\bm{t}^{n-1})$. Consequently, he communicates back the pooled awareness level $\widecheck{\lambda}(\bm{t}^{n-1})$ to all agents.⁶⁶6The current formulation presumes that the mediator, mechanism designer, or social planner has no relevant awareness herself. If she does have relevant awareness, she can incorporate it easily into the message communicated back to agents. Importantly, all agents receive the same message from the mediator. That is, we can use public messages. At the next stage, each agent $i$ must report a type at least at the pooled awareness level $T_{i}^{\widecheck{\lambda}(\bm{t}^{n-1})}$. This report must be consistent with her prior reported type which is implied by the requirement $(t_{i}^{n-1})^{\uparrow}\cap\left(T_{i}^{\widecheck{\lambda}(\bm{t}^{n-1})}\right)^{\uparrow}$. For instance, if an agent reported a type whose awareness is strictly below the pooled awareness of all agents’ reports, then she must now report a type at least at the pooled awareness level $\widecheck{\lambda}(t_{i}^{n-1})$ that is consistent with her prior reported type, i.e., a type in $(t_{i}^{n-1})^{\uparrow}$. However, it could be the case that she has previously pretended to have much less awareness so that the pooled awareness level did not yet incorporate all her awareness. Thus, we allow her to report a type with awareness strictly larger than the pooled awareness level. This motivates the requirement $t_{i}^{n}\in(t_{i}^{n-1})^{\uparrow}\cap\left(T_{i}^{\widecheck{\lambda}(\bm{t}^{n-1})}\right)^{\uparrow}$ rather than $t_{i}^{n}\in(t_{i}^{n-1})^{\uparrow}\cap T_{i}^{\widecheck{\lambda}(\bm{t}^{n-1})}$. If an agent did report at the pooled awareness level in the previous stage, then she is not able to change her report in the current stage. The mechanism stops once no agents revise their reports any further. Once it stops, it implements the physical outcome associated to the final reported type profile by $f_{0}$.

Note that awareness and information is transmitted both from agents to the mediator but also from the mediator to the agents. Thus, awareness of agents may change endogenously when interacting in the mechanism. Since agents report directly types and elaborate in later stages one their prior reported types, we call it a “direct elaboration” mechanism. Note that by the definition of join of the lattice of spaces, $\widecheck{\lambda}(\bm{t}^{n-1})\trianglerighteq\lambda(t_{i}^{n-1})$ for any $i\in I$.

The mechanism stops when no agent wants to further elaborate on her type. Clearly, since $L$ is finite, the mechanism must stop at some finite stage $n$. Moreover, when it stops at $n$, then $\bm{t}^{n}\in\bm{T}^{\widecheck{\lambda}(\bm{t}^{n})}$. That is, all agents must have reported twice in a row types at the same awareness level.

The dynamic direct mechanism induces a game with unawareness in extensive form à la Heifetz et al. (2013a) and Schipper (2021). Compared to standard games in extensive form, extensive-form games with unawareness feature a forest of game trees with exactly one tree for each level of awareness $\ell\in L$. Moreover, the information set of a player at a history in a tree associated with awareness level $\ell$ may be located in a tree associated with a smaller awareness level $\ell^{\prime}\trianglelefteq\ell$. For each awareness level $\ell\in L$, in the corresponding tree, nature moves first drawing for each agent $i$ a payoff type in $T_{i}^{\ell}$ and an awareness level in $L(\ell)$. Across these trees, the draws must be consistent as outlined in the section on payoff types. That is, if payoff type $\bar{t}_{i}\in T^{\bar{\ell}}$ and awareness level $\ell_{i}$ are drawn in the tree associated with $\bar{\ell}$, then payoff type $r^{\bar{\ell}}_{\ell}(\bar{t})\in T^{\ell}$ and awareness level $\ell_{i}\wedge\ell\in L(\ell)$ are drawn in the tree associated with $\ell$. In any game tree of the forest, the move of nature is followed by histories created by the play of the dynamic direct elaboration mechanism. For any $\ell\in L$, the $\ell$-partial game consists of all game trees associated with awareness levels in $L(\ell)$. The idea is that an agent with awareness level $\ell$ can reason about opponents having awareness associated with any level in $L(\ell)$. At any point during the play, the game perceived by the agent with an awareness level $\ell\in L$ is the $\ell$-partial game.

To describe information sets, focus first on initial information sets. If nature draws payoff type $\bar{t}_{i}\in T_{i}^{\bar{\ell}}$ and awareness level $\ell_{i}$ for agent $i$, his initial actual information set is $h^{1}_{i}=r^{\bar{\ell}}_{\ell_{i}}(\bar{t}_{i})$. Here $r^{\bar{\ell}}_{\ell_{i}}(\bar{t}_{i})$ is agent $i$’s initially perceived type. Notice that in this case $h^{1}_{i}$ is an information set in the tree associated with awareness level $\ell_{i}$. That is, in games with unawareness, the information set of a player at a history in a tree may be an object of a “less expressive” tree that misses aspects of which the agent is unaware.

In correspondence to the draw of nature in the full game, for any $\ell\in L$ in the $\ell$-partial game, nature draws payoff type $t_{i}=r^{\bar{\ell}}_{\ell}(\bar{t}_{i})\in T_{i}^{\ell}$ and awareness level $\ell_{i}\wedge\ell\in L(\ell)$ for agent $i$. The initial information set of agent $i$ in the $\ell$-partial game is $h^{1}_{i}=r^{\bar{\ell}}_{\ell_{i}\wedge\ell}(\bar{t}_{i})=r^{\ell}_{\ell_{i}\wedge\ell}(t_{i})$. Here $r^{\ell}_{\ell_{i}\wedge\ell}(t_{i})$ is agent $i$’s initially perceived type in the $\ell$-partial game. (Recall that $\ell_{i}$ is agent $i$’s true initial awareness level selected by nature. It could be that $\ell_{i}\not\trianglelefteq\ell$. In such a case, the meet notation $\ell_{i}\wedge\ell$ becomes crucial when specifying the agent’s awareness level in the $\ell$-partial game as discussed when introducing the payoff type spaces.) Note that even though $t_{i}\in T_{i}^{\ell}$ and thus it is a payoff type in the tree associated with awareness level $\ell$, the corresponding information set and initially perceived payoff type by agent $i$, $h^{1}_{i}=r^{\ell}_{\ell_{i}\wedge\ell}(t_{i})$, is a payoff type in the tree associated with awareness level $\ell_{i}\wedge\ell$ which, by definition of the meet, is necessarily weakly less than $\ell$. Again, in games with unawareness the information set associated to a history in one tree may be an object of a less expressive tree.

Given we have defined all initial information sets, we turn to second stage information sets. For any $i\in I$, fix any initial information set $h_{i}^{1}=t_{i}$. A successor of $h_{i}^{1}$ is defined by $h_{i}^{2}=(t_{i}^{\prime},\bm{t}^{1})$ with $t_{i}^{\prime}\in\left(r_{\lambda(t_{i})}^{\lambda(t_{i})\vee\widecheck{\lambda}(\bm{t}^{1})}\right)^{-1}(t_{i})$ and $\bm{t}^{1}=(t_{i}^{1},\bm{t}_{-i}^{1})$ with $\lambda(t_{i}^{1})\trianglelefteq\lambda(t_{i})$. That is, agent $i$ becoming aware in information set $h_{i}^{2}$ of $\widecheck{\lambda}(\bm{t}^{1})$, joins it with her own awareness $\lambda(t_{i})$, and now considers a more elaborate payoff type $t_{i}^{\prime}\in\left(r_{\lambda(t_{i})}^{\lambda(t_{i})\vee\widecheck{\lambda}(\bm{t}^{1})}\right)^{-1}(t_{i})$. Of course, when communicating at the first stage to the mediator, she could only communicate a payoff type involving awareness not greater than her perceived type at that time $t_{i}$, i.e., $\lambda(t_{i}^{1})\trianglelefteq\lambda(t_{i})$.

Inductively, for any agent $i\in I$ and $n>2$, fix a $n-1$ stage information set $h_{i}^{n-1}=(t_{i},(\bm{t}^{k})_{k<n-1})$. A successor of $h_{i}^{n-1}$ is defined by $h_{i}^{n}=(t_{i}^{\prime},(\bm{t}^{k})_{k<n})$ with $t_{i}^{\prime}\in\left(r_{\lambda(t_{i})}^{\lambda(t_{i})\vee\widecheck{\lambda}(\bm{t}^{n-1})}\right)^{-1}(t_{i})$, $\bm{t}^{n-1}=(t_{i}^{n-1},\bm{t}_{-i}^{n-1})$ with $\lambda(t_{i}^{n-1})\trianglelefteq\lambda(t_{i})$ and $t_{j}^{n-1}\in(t_{j}^{n-2})^{\uparrow}\cap(T_{j}^{\widecheck{\lambda}(\bm{t}^{n-2})})^{\uparrow}$ for all $j\in I$. The interpretations of the conditions are analogous to the ones for $h_{i}^{2}$. The additional condition $t_{j}^{n-1}\in(t_{j}^{n-2})^{\uparrow}\cap(T_{j}^{\widecheck{\lambda}(\bm{t}^{n-2})})^{\uparrow}$ for all $j\in I$ just says that at the previous stage all agents must have reported a (weak) elaboration of the payoff type reported at the stage before the previous stage since this is implied by the dynamic direct elaboration mechanism. We write $h_{i}^{n-1}\rightsquigarrow h_{i}^{n}$ to indicate that $h_{i}^{n}$ is a successor of $h_{i}^{n-1}$.

Since $L$ is finite, the mechanism must stop after some finite number of stages. Denote the set of agent $i$’s information sets by $H_{i}$.

While we defined each information set essentially as a history, most versions of the dynamic direct elaboration mechanisms we will discuss below will not make use of all this information. Our mechanisms will only make use of the awareness embodied in the reported payoff types profiles but not of the precise payoff type profiles reported along the way except for the last one. Yet, writing information sets with sequences of reported payoff type profiles allows us to keep a uniform notation for all mechanisms we discuss and is notationally simpler than alternatives.⁷⁷7A more accurate notation for information sets would be $h_{i}^{n}=(t_{i},(t_{i}^{k},\widecheck{\lambda}(\bm{t}^{k}))_{k<n})$.

We can associate with each information set $h_{i}\in H_{i}$ an awareness level. When $h_{i}^{1}=t_{i}$, agent $i$ has awareness level $\lambda(t_{i})$. This motivates an abuse of notation by letting $\lambda(h^{1}_{i}):=\lambda(t_{i})$ defined by $h_{i}^{1}=t_{i}$. At later stages $n>1$, when $h^{n}_{i}=(t_{i},(\bm{t}^{k})_{k<n})$ occurs, the agent has awareness level $\lambda(t_{i})$. (Recall that by the definition of information sets, $t_{i}$ is already the elaboration of agent $i$’s payoff type in light of awareness raised along the sequence of reported payoff type profiles $(\bm{t}^{k})_{k<n}$.) Again, we abuse notation and write $\lambda(h^{n}_{i}):=\lambda(t_{i})$.

For $h_{i}\in H_{i}$, let $t_{i}(h_{i})=t_{i}$ be defined by $h_{i}^{1}=t_{i}$ if $h_{i}=h^{1}_{i}$ and by $h_{i}^{n}=(t_{i},(\bm{t}^{k})_{k<n})$ if $h_{i}=h^{n}_{i}$. This is the payoff type perceived by agent $i$ in information set $h_{i}$.

Since an agent may not necessarily be aware of everything ex-ante, she cannot necessarily envision all of her information sets and choose a strategy that prescribes an action to any of her information sets. At any information set $h_{i}\in H_{i}$, agent $i$’s perception of the game is the $\lambda(h_{i})$-partial game. For any $\ell\in L$, the set of information sets perceived by her in the $\ell$-partial game is $H_{i}^{\ell}:=\{h_{i}\in H_{i}:\lambda(h_{i})\trianglelefteq\ell\}$.

Having specified the representation of information sets in the game with unawareness induced by the dynamic direct elaboration mechanism, we proceed with defining strategies. As standard in game theory, a strategy of an agent assigns to each of her information sets an action. More formally, a (pure) strategy of agent $i$ in the game induced by the dynamic direct elaboration mechanism is a mapping $\sigma_{i}:H_{i}\longrightarrow\mathcal{T}_{i}$ such that

• (i)

  for all $h^{1}_{i}\in H_{i}$, $\sigma_{i}(h^{1}_{i})\in\bigcup_{\ell\in L(\lambda(h_{i}))}T_{i}^{\ell}$,

• (ii)

  for $n\in\mathbb{N}$ and $h_{i}^{n}\in H_{i}$, $\sigma_{i}(h_{i}^{n})\in(t_{i}^{n-1})^{\uparrow}\cap(T_{i}^{\lambda(h^{n}_{i})})^{\uparrow}$, where $t_{i}^{n-1}$ is agent $i$’s previous period’s reported payoff type according to $h_{i}^{n}$.

Property (i) applies to all initial information sets of agent $i$. It says that the agent can report any type that she is aware of at her perceived true type. Property (ii) is a constraint imposed by the dynamic direct elaboration mechanism as it allows only elaborations of the previously reported type.⁸⁸8This constraint is not necessary for our results. However, it is a natural simplifying assumption on communication, reducing the set of messages over which each agent optimizes and reducing the size of game trees. An agent can only provide elaborations of her payoff type that she is aware of herself at the information set. She has to elaborate at least at the pooled awareness level $\ell$. Yet, she is also free to elaborate at an even greater awareness level.

For any awareness level $\ell\in L$ and strategy $\sigma_{i}$, an $\ell$-partial strategy $\sigma_{i}^{\ell}$ is the strategy $\sigma_{i}$ restricted to information sets in $H_{i}^{\ell}$. It assigns an action to each information set of agent $i$ in any game tree of the $\ell$-partial game induced by the mechanism.

We denote by $\Sigma_{i}$ the set of agent $i$’s strategies and by $\Sigma_{i}^{\ell}$ the set of agent $i$’s $\ell$-partial strategies for $\ell\in L$. We denote by $\bm{\Sigma}_{-i}:=\times_{j\in I\setminus\{i\}}\Sigma_{j}$, $\bm{\Sigma}_{-i}^{\ell}:=\times_{j\in I\setminus\{i\}}\Sigma_{j}^{\ell}$, $\bm{\Sigma}:=\times_{i\in I}\Sigma_{i}$ and $\bm{\Sigma}^{\ell}:=\times_{i\in I}\Sigma_{i}^{\ell}$ the set of strategy profiles of agent $i$’s opponents, the set of $\ell$-partial strategy profiles of agent $i$’s opponents, the set of strategy profiles, and the set of $\ell$-partial strategy profiles, respectively.

For agent $i$, the truth-telling and elaboration strategy $\sigma^{*}_{i}$ is defined by:

• •

  For any $h_{i}^{1}\in H_{i}$, $h^{1}_{i}=t_{i}$ implies $\sigma_{i}^{*}(h_{i}^{1})=t_{i}$.

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  For any $n>1$, $h_{i}^{n}\in H_{i}$, $h_{i}^{n}=(t_{i},(\bm{t}^{k})_{k<n})$ implies $\sigma_{i}^{*}(h_{i}^{n})=t_{i}$.

That is, at each information set, agent $i$ tells her true payoff type as currently perceived at this information set. (Note that at $h^{n}_{i}=(t_{i},(\bm{t}^{k})_{k<n})$, payoff type $t_{i}$ incorporates her awareness associated with $\bm{t}^{n-1}$.)

For any agent $i\in I$, awareness level $\ell\in L$, “initial” payoff type profile $\bm{t}=(t_{j})_{j\in I}\in\bm{T}^{\ell}$, “initial” profile of awareness levels $\bm{\ell}=(\ell_{j})_{j\in I}\in L(\ell)^{|I|}$, and profile of strategies $\bm{\sigma}=(\sigma_{j})_{j\in I}\in\bm{\Sigma}$, we let $H_{i}(\bm{t},\bm{\ell},\bm{\sigma})$ denote the set of agent $i$’s histories reached with $\bm{t}$, $\bm{\ell}$, and $\bm{\sigma}$ (in the $\ell$-partial game). It is defined inductively as follows:

The initial information set is agent $i$’s perceived payoff type in the $\ell$-partial game. At successive information sets, the initial payoff type is elaborated in light of the pooled awareness raised in the reports induced by the strategy profile at their prior reached information sets. The elaboration is consistent with the initial payoff type profile selected in $\bm{T}^{\ell}$. This feature implies that $H_{i}(\bm{t},\bm{\ell},\bm{\sigma})$ consists of a unique path (a unique sequence) of information sets for agent $i$. The requirement $h_{i}^{n-1}\rightsquigarrow h_{i}^{n}$ implies that the sequence $(\bm{t}^{k})_{k<n-1}$ in the definition of $h_{i}^{n}$ is identical to the sequence of payoff type profiles in $h_{i}^{n-1}$.

For any agent $i\in I$, awareness level $\ell\in L$, “initial” payoff type profile $\bm{t}=(t_{j})_{j\in I}\in\bm{T}^{\ell}$, “initial” profile of awareness levels $\bm{\ell}=(\ell_{j})_{j\in I}\in L(\ell)^{|I|}$, and strategy $\sigma_{i}\in\Sigma_{i}$, we let $H_{i}(\bm{t},\bm{\ell},\sigma_{i}):=\bigcup_{\bm{\sigma}_{-i}\in\bm{\Sigma}_{-i}}H_{i}(\bm{t},\bm{\ell},\sigma_{i},\bm{\sigma}_{-i})$. Similarly, we let $H_{i}(\sigma_{i}):=\bigcup_{(\bm{t},\bm{\ell})\in\bigcup_{\ell\in L}\bm{T}^{\ell}\times L(\ell)^{|I|}}H_{i}(\bm{t},\bm{\ell},\sigma_{i})$.

For every $\ell\in L$, any “initial” payoff type profile $\bm{t}=(t_{j})_{j\in I}\in\bm{T}^{\ell}$, “initial” profile of awareness levels $\bm{\ell}=(\ell_{j})_{j\in I}\in L(\ell)^{|I|}$, and profile of strategies $\bm{\sigma}=(\sigma_{j})_{j\in I}\in\bm{\Sigma}$ give rise to a unique sequence of payoff type profiles $\tau(\bm{t},\bm{\ell},\bm{\sigma})=(\bm{t}^{1},...,\bm{t}^{n-1},\bm{t}^{n})$ (for some $n>2$) defined by $\bm{t}^{k}=(\sigma_{j}(h_{j}^{k}))_{j\in I}$ with $h_{j}^{k}\in H_{j}(\bm{t},\bm{\ell},\bm{\sigma})$ for $j\in I$, $k\leq n$, and $\bm{t}^{n-1}=\bm{t}^{n}$. This sequence is unique because as we observed above $h_{j}^{1}$ is unique for each $j\in I$ given $\bm{t},\bm{\ell}$, and $\bm{\sigma}$, $h_{j}^{2}$ is unique for each $j\in I$ given $\bm{t},\bm{\ell}$, and $\bm{\sigma}$, etc. The requirement $\bm{t}^{n-1}=\bm{t}^{n}$ means that $\tau(\bm{t},\bm{\ell},\bm{\sigma})$ consists of the entire sequence of payoff type profiles until the dynamic direct elaboration mechanisms stops. For the implementation of outcomes, we are especially interested in the last profile of payoff types of any such sequence and denote it by $\tau^{*}(\bm{t},\bm{\ell},\bm{\sigma})$. The set of all sequences is denoted by $\mathfrak{T}:=\{\tau(\bm{t},\bm{\ell},\bm{\sigma}):\bm{\sigma}\in\bm{\Sigma},(\bm{t},\bm{\ell})\in\bigcup_{\ell\in L}(\bm{T}^{\ell}\times L(\ell)^{|I|})\}$

For any agent $i\in I$ and information set $h_{i}\in H_{i}$, define

This is the set of agent $i$’s strategies that are consistent with information set $h_{i}$.

For any agent $i\in I$, let $f_{i}:\mathfrak{T}\longrightarrow\mathbb{R}$ denote the transfer paid to agent $i$ in the dynamic direct elaboration mechanisms. These transfers will depend on the precise versions of the direct elaboration mechanism studied below. In contrast to the outcome function $f_{0}$ we allow transfers to depend on the entire sequence of reported type profiles. The reason is that it will become important who first reported which awareness level.

We let the social choice function (i.e., the outcome function and transfers) be denoted by $f:\mathfrak{T}\longrightarrow\mathbb{R}$ and defined by $f=(f_{0},f_{1},...,f_{|I|})$. That is, for any strategy profile $\bm{\sigma}\in\bm{\Sigma}$, $\ell\in L$, initial profiles of payoff types $\bm{t}\in\bm{T}^{\ell}$, initial profiles of awareness levels $\bm{\ell}\in L(\ell)^{|I|}$, $f(\tau(\bm{t},\bm{\ell},\bm{\sigma}))=\left(f_{0}(\tau^{*}(\bm{t},\bm{\ell},\bm{\sigma})),f_{1}(\tau(\bm{t},\bm{\ell},\bm{\sigma})),...,f_{|I|}(\tau(\bm{t},\bm{\ell},\bm{\sigma}))\right)$.

Conditional dominance strengthens dominance by requiring each agent not only to select a (partial) strategy that is ex-ante dominant but also dominant conditional on each information set. This becomes important when agents cannot anticipate all information sets ex-ante and thus cannot select ex-ante a strategy for the entire game in extensive form.

Recall that $\left(\bm{\bar{t}},(\ell_{i})_{i\in I}\right)\in\bm{T}^{\bar{\ell}}\times L^{|I|}$ represents the move of nature selecting both actual payoff types and awareness levels for all agents. Consequently, $\bigvee_{i\in I}\ell_{i}$ represents the pooled awareness level.