<p>I agree with Michael Greinecker: I cannot find useful necessary and sufficient conditions for a Nash equilibrium to exist.</p>
<p>A difficulty is that auctions with common values and complete information are often discontinuous games --- a bidder's utility jumps when her bid increases to the point where it is no longer the losing bid.
Such discontinuities preclude the usage of the Glicksberg-Fan theorem to prove the existence of an equilibrium. This is Proposition 8.D.3 in Mas-Colell's Microeconomic Theory.</p>
<p><a href="https://www.annualreviews.org/content/journals/10.1146/annurev-economics-082019-111720" rel="nofollow noreferrer">Reny (2020)</a> surveys existing existence theorems in discontinuous games. Theorems 3-11 provide an inventory of sufficient conditions for an equilibrium to exist.</p>

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2024-08-31T15:39:17.060Z
2024-08-31T15:39:17.060Z
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<p>Suppose that a seller wants to allocate a single object among <span class="math-container">$n \in \{2, 3, \dots\}$</span> agents.
Agents know the object is worth <span class="math-container">$v \geq 0$</span>, so that values are common and agents have no private information.</p>
<p>An auction is a tuple <span class="math-container">$(M_1, ..., M_n, q, t)$</span>.
<span class="math-container">$M_i$</span> is the set of messages available to agent <span class="math-container">$i$</span> and <span class="math-container">$M = M_1 \times \dots \times M_n$</span> is the set of message profiles.
<span class="math-container">$q = (q_1, \dots, q_n)$</span> is the allocation rule, which specifies a probability to each agent receiving the object for each profile of messages, so that <span class="math-container">$q_i: M \mapsto [0, 1]$</span> and <span class="math-container">$\sum_{i=1}^n q_i(m) \leq 1$</span> for all <span class="math-container">$m \in M$</span>.
<span class="math-container">$t = (t_1, \dots, t_n)$</span> is the transfer rule that determines how much each agent must pay the seller for each profile of messages, so that <span class="math-container">$t_i: M \mapsto \mathbb{R}$</span>.</p>
<p>Agent <span class="math-container">$i$</span>'s payoff at message profile <span class="math-container">$m$</span> is: <span class="math-container">$u_i(m) = q_i(m) v - t_i(m)$</span>.
A strategy for <span class="math-container">$i$</span> is a probability measure <span class="math-container">$\sigma_i$</span> on <span class="math-container">$M_i$</span>.
<span class="math-container">$i$</span>'s expected payoff at strategy profile <span class="math-container">$\sigma = (\sigma_1, \dots, \sigma_n)$</span> is <span class="math-container">$\overline{u}_i(\sigma) = \int_{M} u_i(m) \sigma_1(dm_1) \cdots \sigma_n(dm_n)$</span>.
A Nash equilibrium is a profile of strategies <span class="math-container">$\sigma$</span> such that <span class="math-container">$\overline{u}_i(\sigma) \geq \overline{u}_i(\sigma_i', \sigma_{-i})$</span> for all <span class="math-container">$i$</span> and for every alternative strategy <span class="math-container">$\sigma_i'$</span>, where <span class="math-container">$\sigma_{-i}$</span> refers to the equilibrium strategies of agents other than <span class="math-container">$i$</span>.</p>
<p>I restrict attention to auctions where every agent can send a special message <span class="math-container">$0$</span> that ensures a non-negative payoff: <span class="math-container">$0 \in M_i$</span> and <span class="math-container">$q_i(0, m_{-i}) v - t_i(0, m_{-i}) \geq 0$</span> for all <span class="math-container">$i$</span> and for all messages of others <span class="math-container">$m_{-i}$</span>. This property is known as participation security (Brooks and Du 2021).</p>
<p><strong>Question:</strong> are there known necessary and sufficient conditions on <span class="math-container">$(M_1, \dots, M_n, q, t)$</span> for a Nash equilibrium to exist? If so, what are they?</p>
<p>I know of the following sufficient conditions for equilibrium existence:</p>
<ol>
<li><span class="math-container">$M_1, \dots, M_n$</span> are all finite. This follows from Nash's existence theorem (Jehle and Reny, Theorem 7.2)</li>
<li>(i) <span class="math-container">$M_i \subset \mathbb{R}^k$</span> is nonempty, convex and compact for all <span class="math-container">$i$</span>, and (ii) <span class="math-container">$q_i(m_1, \dots, m_n) v - t_i(m_1, \dots, m_n)$</span> is a continuous function that is quasiconcave in <span class="math-container">$m_i$</span> (Mas-Colell, Proposition 8.D.3).</li>
</ol>
<p>Anecdotally, the first-price, second-price and English auctions also admit an equilibrium.
In the first- and second-price auctions, there is an equilibrium where every agent bids <span class="math-container">$v$</span>.
In the English auction, there is an equilibrium where all agents drop out at <span class="math-container">$v$</span>.</p>

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2024-08-27T03:45:21.293Z
[Milgrom and Weber (1982)](https://www.econometricsociety.org/publications/econometrica/1982/09/01/theory-auctions-and-competitive-bidding) is probably closest to what you're trying to find. While it doesn't provide necessary conditions on $(M,q,t)$ as such, it does list sufficient conditions guaranteeing the existence of (at least one) BNE in various forms of auctions with affiliated values (common value auction being a special case thereof).
Herr K.
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2024-08-28T13:10:52.303Z
If $M$ is nonempty and compact metrizable and $q$ and $t$ are continuous, an equilib rium (in possibly mixed strategies) exists by the Glicksberg-Fan existence theorem. I doubt there are interesting conditions that are necessary and sufficient, but Tian's work on transfer-continuity might be relevant to that question.
Michael Greinecker
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2024-08-28T15:19:24.700Z
@HerrK. Thanks! Milgrom and Weber (1982) showcase equilibria in the second-price, first-price and English auction formats, with and without reserve prices, in the affiliated values model. This model does include the case of common values and complete information. But I'm wondering about existence across selling mechanisms (in this very simple information structure that features common values and complete information).
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Existence of Nash equilibrium in auctions with common values and complete information
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# Existence of Nash equilibrium in auctions with common values and complete information

(Asked by: username_0 on 2024-08-21)

Suppose that a seller wants to allocate a single object among $n \in \{2, 3, \dots\}$ agents.
Agents know the object is worth $v \geq 0$, so that values are common and agents have no private information.

An auction is a tuple $(M\_1, ..., M\_n, q, t)$.
$M\_i$ is the set of messages available to agent $i$ and $M = M\_1 \times \dots \times M\_n$ is the set of message profiles.
$q = (q\_1, \dots, q\_n)$ is the allocation rule, which specifies a probability to each agent receiving the object for each profile of messages, so that $q\_i: M \mapsto [0, 1]$ and $\sum\_{i=1}^n q\_i(m) \leq 1$ for all $m \in M$.
$t = (t\_1, \dots, t\_n)$ is the transfer rule that determines how much each agent must pay the seller for each profile of messages, so that $t\_i: M \mapsto \mathbb{R}$.

Agent $i$'s payoff at message profile $m$ is: $u\_i(m) = q\_i(m) v - t\_i(m)$.
A strategy for $i$ is a probability measure $\sigma\_i$ on $M\_i$.
$i$'s expected payoff at strategy profile $\sigma = (\sigma\_1, \dots, \sigma\_n)$ is $\overline{u}\_i(\sigma) = \int\_{M} u\_i(m) \sigma\_1(dm\_1) \cdots \sigma\_n(dm\_n)$.
A Nash equilibrium is a profile of strategies $\sigma$ such that $\overline{u}\_i(\sigma) \geq \overline{u}\_i(\sigma\_i', \sigma\_{-i})$ for all $i$ and for every alternative strategy $\sigma\_i'$, where $\sigma\_{-i}$ refers to the equilibrium strategies of agents other than $i$.

I restrict attention to auctions where every agent can send a special message $0$ that ensures a non-negative payoff: $0 \in M\_i$ and $q\_i(0, m\_{-i}) v - t\_i(0, m\_{-i}) \geq 0$ for all $i$ and for all messages of others $m\_{-i}$. This property is known as participation security (Brooks and Du 2021).

**Question:** are there known necessary and sufficient conditions on $(M\_1, \dots, M\_n, q, t)$ for a Nash equilibrium to exist? If so, what are they?

I know of the following sufficient conditions for equilibrium existence:

1. $M\_1, \dots, M\_n$ are all finite. This follows from Nash's existence theorem (Jehle and Reny, Theorem 7.2)
2. (i) $M\_i \subset \mathbb{R}^k$ is nonempty, convex and compact for all $i$, and (ii) $q\_i(m\_1, \dots, m\_n) v - t\_i(m\_1, \dots, m\_n)$ is a continuous function that is quasiconcave in $m\_i$ (Mas-Colell, Proposition 8.D.3).

Anecdotally, the first-price, second-price and English auctions also admit an equilibrium.
In the first- and second-price auctions, there is an equilibrium where every agent bids $v$.
In the English auction, there is an equilibrium where all agents drop out at $v$.

**Tags:** game-theory, nash-equilibrium, auctions, mechanism-design, information
> **Comments:**
> **username_1**: [Milgrom and Weber (1982)](https://www.econometricsociety.org/publications/econometrica/1982/09/01/theory-auctions-and-competitive-bidding) is probably closest to what you're trying to find. While it doesn't provide necessary conditions on $(M,q,t)$ as such, it does list sufficient conditions guaranteeing the existence of (at least one) BNE in various forms of auctions with affiliated values (common value auction being a special case thereof).  
> **username_2**: If $M$ is nonempty and compact metrizable and $q$ and $t$ are continuous, an equilib rium (in possibly mixed strategies) exists by the Glicksberg-Fan existence theorem. I doubt there are interesting conditions that are necessary and sufficient, but Tian's work on transfer-continuity might be relevant to that question.  
> **username_0**: @HerrK. Thanks! Milgrom and Weber (1982) showcase equilibria in the second-price, first-price and English auction formats, with and without reserve prices, in the affiliated values model. This model does include the case of common values and complete information. But I'm wondering about existence across selling mechanisms (in this very simple information structure that features common values and complete information).  

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## 1 answers

### Answer by username_0 on 2024-08-31. Score: 1 | Accepted answer

I agree with Michael Greinecker: I cannot find useful necessary and sufficient conditions for a Nash equilibrium to exist.

A difficulty is that auctions with common values and complete information are often discontinuous games --- a bidder's utility jumps when her bid increases to the point where it is no longer the losing bid.
Such discontinuities preclude the usage of the Glicksberg-Fan theorem to prove the existence of an equilibrium. This is Proposition 8.D.3 in Mas-Colell's Microeconomic Theory.

[Reny (2020)](https://www.annualreviews.org/content/journals/10.1146/annurev-economics-082019-111720) surveys existing existence theorems in discontinuous games. Theorems 3-11 provide an inventory of sufficient conditions for an equilibrium to exist.


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