---
title: S-Supporting Equilibria in Game Theory
url: https://www.emergentmind.com/topics/s-supporting-equilibria
type: topic
---

# S-Supporting Equilibria in Game Theory

Searching arXiv for the cited papers and closely related work on S-supporting equilibria across game-theoretic contexts.
“S-supporting equilibria” is a context-dependent expression in contemporary game theory. In one line of work on bimatrix games, an $\epsilon$-well-supported Nash equilibrium is called $S$-supporting when each player’s mixed strategy has support of cardinality at most $S$ [1504.03602]. In market design, an equilibrium is “$x^*$-supporting” when anonymous prices and agent-specific budgets support a prescribed Pareto-optimal allocation $x^*$ [2103.08634]. In repeated additive games with reactive strategies, non-empty subsets $S$ of the action set index equilibrium classes in which self-play uses exactly the actions in $S$ [2606.27653]. A further usage studies which subset of players must be fixed at a target equilibrium so that all remaining players best-respond to it [2312.03229]. This suggests that the expression is not a single standardized definition but a family of related notions organized around what is being “supported”: a bounded mixed support, a target allocation, a self-play action set, or an equilibrium profile itself.

## 1. Terminological scope

The main usages can be organized as follows.

| Context | Meaning of “supporting” | Equilibrium object |
|---|---|---|
| Win-lose bimatrix games | Both mixed supports have cardinality $\le S$ | $\epsilon$-WSNE |
| Competitive equilibrium with unequal budgets | Prices and budgets support a target allocation $x^*$ | $(p,b,x^*)$ |
| Repeated additive games | Self-play uses exactly the actions in $S$ | Symmetric reactive NE class |
| Strategic-form control | A player subset makes the target equilibrium a best response for everyone else | Direct control set |

In bimatrix approximation, the relevant object is the support of a mixed strategy: if $(p,q)$ is an $\epsilon$-well-supported Nash equilibrium, then “$S$-supporting” means that both $\mathrm{supp}(p)$ and $\mathrm{supp}(q)$ have cardinality at most $S$ [1504.03602]. In Fisher-style markets with divisible goods, the central question is instead whether one can choose anonymous prices and agent-specific budgets so that a given feasible allocation $x^*$ is individually optimal and market-clearing [2103.08634]. In repeated additive games with reactive strategies, $S$ is a subset of the action set, and an $S$-supporting equilibrium is one whose stationary self-play distribution is supported on $S\times S$ [2606.27653]. In direct-control models, “supporting” refers neither to mixed-strategy support size nor to action subsets, but to a coalition of players whose forced compliance makes the target equilibrium a best response for the remaining players [2312.03229].

A common source of confusion is therefore terminological rather than mathematical. Results about $S$-supporting $\epsilon$-WSNE do not transfer directly to $x^*$-supporting competitive equilibria or to $S$-supporting reactive equilibria, because the underlying state spaces, feasibility constraints, and equilibrium predicates are different.

## 2. $S$-supporting $\epsilon$-well-supported Nash equilibria in win-lose games

For a bimatrix game $(A,B)$ with $A,B\in[0,1]^{m\times n}$, a pair of mixed strategies $p\in\Delta_m$, $q\in\Delta_n$ is an $\epsilon$-well-supported Nash equilibrium if every pure strategy played with positive probability is within $\epsilon$ of a best response. Equivalently,
\[
e_i^T A q \ge e_j^T A q - \epsilon
\]
for all $i$ with $p_i>0$ and all $j=1,\dots,m$, and
\[
p^T B e_j \ge p^T B e_i - \epsilon
\]
for all $j$ with $q_j>0$ and all $i=1,\dots,n$. Such an $\epsilon$-WSNE is $S$-supporting if both supports have cardinality at most $S$ [1504.03602].

For win-lose games, every instance can be encoded as a directed bipartite graph $G=(R\cup C,E)$ with row vertices $R=\{r_1,\dots,r_m\}$ and column vertices $C=\{c_1,\dots,c_n\}$, where
\[
(r_i,c_j)\in E \iff A_{ij}=1,\qquad (c_j,r_i)\in E \iff B_{ij}=1.
\]
Assuming every vertex has out-degree at least $1$, the central characterization states that for any integer $k\ge 1$ and any $\epsilon$ with $1-\epsilon>0$, the game admits an $\epsilon$-WSNE with both supports of cardinality at most $k$ if and only if the graph contains either a directed cycle of length at most $2k$, or an undominated set of $k$ vertices all on one side of the bipartition [1504.03602].

The characterization isolates two distinct mechanisms for small-support well-supported equilibria. A short directed cycle supplies a compact alternating structure of mutually rewarding actions. An undominated set supplies a side of the game for which no single opponent action covers all supported actions simultaneously. The proof decomposes accordingly: one lemma constructs an equilibrium from an undominated set, another from a short directed cycle, and the converse shows that any $\epsilon$-WSNE with small supports forces one of those two combinatorial configurations. In the converse direction, if neither support set is undominated, then both players have true best responses of payoff $1$ against the opponent’s mix; in the induced subgraph on the supports, every vertex has out-degree at least $1$, hence there is a directed cycle of length at most $|P\cup Q|\le 2k$ [1504.03602].

This graph-theoretic reformulation is significant because it converts a mixed-strategy existence question into a finite structural question about bipartite digraphs. In the win-lose setting, support-bounded $\epsilon$-WSNE are therefore governed by cycle structure and domination structure rather than by generic fixed-point arguments.

## 3. Lower bounds and barriers for small supports

The main lower bound for well-supported equilibria is negative in the strongest possible constant-support sense: for every fixed $k$ and any $\epsilon<1$, there are win-lose bimatrix games for which every $\epsilon$-WSNE has support size greater than $k$ [1504.03602]. The proof uses additive number theory. Haight’s theorem yields, for any $k$, a modulus $q$ and a subset $Y\subseteq\mathbb{Z}_q$ with $Y-Y=\mathbb{Z}_q$ and $0\notin (k)Y$. From this, one constructs a non-bipartite digraph of girth at least $k$ in which every pair of vertices is dominated by some third vertex, and then converts it into a bipartite digraph by doubling vertices and adding self-arcs. The resulting win-lose game has no cycle of length at most $2k$ and every set of $k$ rows or columns is dominated, so the characterization theorem excludes every $\epsilon$-WSNE with supports of size at most $k$ [1504.03602].

This construction has two further consequences. First, it shows that the minimum support required for $\epsilon$-WSNE is unbounded as $k$ grows, even though the argument is existential rather than quantitative in terms of elementary closed forms. Second, it refutes graph-theoretic conjectures of Daskalakis, Mehta and Papadimitriou, and of Myers, by producing $(k,l)$-digraphs for all $k,l$ rather than forcing the conjectured dichotomy between short cycles and small undominated sets [1504.03602].

A complementary barrier was established earlier for the regime $\epsilon<2/3$. In win-lose games on $n$ strategies, every $\epsilon$-WSNE may require each player’s support to have size at least $\Omega((\log n)^{1/3})$ [1309.7258]. The construction uses a bipartite digraph derived from a random tournament and a “$k$-covered” property ensuring that every $k$-set on either side has a common in-neighbour. The contradiction argument combines this covering property with a bipartite Caccetta-Häggkvist-type theorem: if every vertex in a balanced bipartite digraph on $k+k$ vertices has in-degree greater than $k/3$, then the graph contains a directed $4$-cycle. Since the constructed game has no directed $2$- or $4$-cycles, any $\epsilon$-WSNE with $\epsilon<2/3$ must have larger supports [1309.7258].

The same paper also supplies the positive side of the support-size trade-off: in every bimatrix game with payoffs in $[0,1]$, for any $\epsilon>0$ there is an $\epsilon$-WSNE in which each player mixes uniformly on a multiset of
\[
t=\Bigl\lceil \frac{2}{\epsilon^2}(\ln m+\ln 2)\Bigr\rceil
=O\!\left(\frac{\log n}{\epsilon^2}\right)
\]
pure strategies [1309.7258]. Thus the lower bound is polylogarithmic and the upper bound is polylogarithmic, though with different exponents and under different parameter regimes. The paper also proves a sharp “no size-$2$” phenomenon: for every $\delta>0$, there exist win-lose games for which no pair of mixed strategies with supports of size at most two forms a $(1-\delta)$-WSNE, even though every bimatrix game with payoffs in $[0,1]$ admits a $1/2$-approximate Nash equilibrium with supports of cardinality at most two [1309.7258]. This is one of the clearest separations between approximate Nash equilibrium and its well-supported variant.

## 4. Small-support equilibria beyond well-supported Nash

The small-support perspective extends well beyond $\epsilon$-WSNE. For $n$-player games with $m$ actions per player, every game admits an $\epsilon$-approximate Nash equilibrium in which each player mixes over at most
\[
S=\left\lceil \frac{16\,(n-1)\,(\ln n+\ln m)}{\epsilon^2}\right\rceil
\]
pure actions [1305.2432]. Babichenko’s proof uses population-splitting and Lipschitz smoothing: each original player is replaced by a population of $k$ subplayers, the smoothed game becomes $1/k$-Lipschitz in every opponent’s action, and concentration for Lipschitz functions then implies the existence of a pure equilibrium in the population game, which corresponds to a $k$-uniform $\epsilon$-equilibrium in the original game. For graphical games of maximum degree $d$, the same argument yields
\[
S=\left\lceil \frac{16\,d\,(\ln n+\ln m)}{\epsilon^2}\right\rceil,
\]
so the dependence shifts from the total number of players to the interaction degree [1305.2432].

Approximate correlated equilibrium and approximate coarse correlated equilibrium admit analogous polylogarithmic support bounds. In every $n$-player $m$-action game and for every $\epsilon>0$, there exists an $\epsilon$-CCE with support size at most
\[
k \ge \frac{2(\ln m+\ln n+\ln(1/\delta))}{\epsilon^2},
\]
and there exists an $\epsilon$-CE with support size at most
\[
k \ge \frac{264\,\ln m\,(\ln m+\ln n-\ln \epsilon+\ln(16/\delta))}{\epsilon^4}
\]
for any chosen failure probability $\delta$ [1308.6025]. The probabilistic method behind both results is similar: start from an exact equilibrium, sample profiles independently, and apply Hoeffding’s inequality plus a union bound over regret constraints. For $\epsilon$-CCE, this leads to a randomized algorithm with expected number of iterations at most $2$ once an exact correlated equilibrium is available; each iteration checks the $nm$ deviation inequalities in $O(knm)$ time [1308.6025].

These results show that “small support” is a robust theme across equilibrium notions, but not a uniform one. Approximate Nash, $\epsilon$-WSNE, $\epsilon$-CE, and $\epsilon$-CCE impose different deviation classes, and the resulting support-size bounds differ accordingly. The support-minimization problem also changes with the equilibrium concept: finding an exact correlated equilibrium of minimum support is NP-hard under Cook reductions even in two-player zero-sum games [1308.6025].

## 5. Supporting arbitrary Pareto-optimal allocations in competitive equilibrium

In a market with additive valuations over heterogeneous divisible resources, an allocation $x^*$ is “supported” when there exist anonymous prices $p\ge 0$ and agent-specific budgets $b\ge 0$ such that $(p,b,x^*)$ is a competitive equilibrium [2103.08634]. The model has agents $N=\{1,2,\dots,n\}$, goods $M=\{1,2,\dots,m\}$ with supplies $s\in\mathbb{R}^m_{>0}$, and additive utilities
\[
v_i(x_i)=\sum_{j\in M} v_{ij}x_{ij}.
\]
The tuple $(p,b,x^*)$ is $x^*$-supporting if: each agent solves
\[
\max_{x_i\ge 0,\; p\cdot x_i\le b_i} v_i(x_i),
\]
no agent overspends, and market clearing holds:
\[
\sum_{i=1}^n x^*_{ij}=s_j\qquad \forall j\in M.
\]

The core theorem states that given any Pareto-optimal allocation $x^*$, one can compute in polynomial time prices $p\ge 0$ and budgets $b\ge 0$ so that $(p,b,x^*)$ is an equilibrium supporting $x^*$ [2103.08634]. The construction has three phases.

First, a cycle-elimination procedure transforms the sharing graph of the allocation into a forest while preserving Pareto optimality and each agent’s utility. If the bipartite sharing graph $G(y)$ contains a cycle, an infinitesimal reallocation around that cycle can eliminate one strictly interior edge without decreasing any agent’s utility. Repeating this produces a new Pareto-optimal allocation $x$ with the same utility vector and a cycle-free sharing graph [2103.08634].

Second, each tree component is priced locally. Choosing an arbitrary agent $r$ as root, prices on adjacent items are initialized by $p_j\leftarrow v_{rj}$, and then propagated through the tree by
\[
p_k \leftarrow v_{ik}\cdot p_{j'}/v_{i,j'}
\]
whenever agent $i$ holds both the already priced item $j'$ and another item $k$. This enforces indifference among all goods held by the same agent within the tree:
\[
v_{i,k}/p_k=v_{i,k'}/p_{k'}.
\]
Each agent budget is then set to exactly finance the assigned bundle:
\[
b_i=\sum_{j\in T} p_j x_{ij}.
\]
Lemmas 3.3–3.4 show that under these local prices and budgets, no agent in the tree can profitably reallocate budget from one good in the tree to another [2103.08634].

Third, the locally priced trees are scaled to prevent cross-tree deviations. One chooses a scalar $\alpha_T>0$ per tree and sets
\[
p_j\leftarrow \alpha_{T(j)}p_j^T,\qquad b_i\leftarrow \alpha_{T(i)}b_i^T.
\]
A continuous gain map $F$ is defined on the simplex of scaling vectors, and Brouwer’s fixed-point theorem yields $\alpha^*$ with $F(\alpha^*)=\alpha^*$. At such a fixed point, all cross-tree gains vanish. Equivalently, the paper gives a linear-program formulation:
\[
\forall i,j:\quad (u_i/b_i^{T(i)})\cdot \alpha_{T(j)} \ge (v_{ij}/p_j^{T(j)})\cdot \alpha_{T(i)},
\]
together with $\sum_T \alpha_T=1$ and $\alpha_T\ge \lambda>0$ [2103.08634].

Algorithmically, cycle elimination can be done in $O(\#\text{edges}\cdot \mathrm{poly}(1))$, pricing each tree is a single BFS or DFS in $O(|E|)$, and the scaling step can be solved in polynomial time by LP. The same framework supports the Rawlsian max-min allocation, which can itself be found by a single LP in $O(n\cdot m\cdot \mathrm{polylog})$, or by sorting in $O(m\log m)$ for two agents and in $O(n\log n)$ for two goods [2103.08634]. In this literature, “supporting equilibrium” means implementability of a target Pareto-optimal allocation rather than sparsity of a mixed strategy.

## 6. $S$-supporting classes in repeated additive games

In repeated additive games between two players with finitely many actions, Lesigang, Hilbe and Glynatsi study reactive strategies, i.e. memory-1 strategies that condition only on the opponent’s previous action [2606.27653]. The stage game is additive:
\[
g(a,b)=g_A(a)+g_B(b),
\]
and the long-run payoff under reactive strategies exists as
\[
\pi^i(p,\tilde p)=\lim_{\tau\to\infty}\frac1\tau\sum_{t=1}^\tau \pi_t^i(p,\tilde p).
\]
A reactive strategy satisfies $p_{a_1,b}=p_{a_2,b}$ for all $a_1,a_2\in A$, so it can be written as $\{p_b\}_{b\in B}$ with each $p_b\in \Delta(A)$ [2606.27653].

A symmetric Nash equilibrium in the class of reactive strategies is a $p$ such that
\[
\pi^i(p,p)\ge \pi^i(q,p)\qquad \forall q\text{ reactive strategy.}
\]
A key simplification is that one only needs to check deviations among pure unconditional strategies $\mathrm{All}\,a$:
\[
p\text{ is NE}\quad\Longleftrightarrow\quad \pi^i(p,p)\ge \pi^i(\mathrm{All}\,a,p)\quad \forall a\in A.
\]
Moreover, if $v(a,\bar a)$ is the stationary frequency of $(a,\bar a)$ under self-play and
\[
\mu(a)=\sum_{\tilde a\in A} v(a,\tilde a),
\]
then
\[
\pi^i(p,p)=\sum_{a\in A}\mu(a)\,\pi^i(\mathrm{All}\,a,p).
\]
Because
\[
\pi^i(\mathrm{All}\,a,p)=g_A^i(a)+\sum_{\tilde a\in A} g_B^i(\tilde a)\,p_a(\tilde a),
\]
the equilibrium conditions become linear equalities and inequalities in the strategy variables [2606.27653].

Theorem 3.1 states that a reactive strategy $p$ is a symmetric Nash equilibrium if and only if there exists a non-empty $S\subseteq A$ such that four conditions hold. First, a support condition:
\[
p_b(a)=0\qquad \forall\, b\in S,\ a\notin S.
\]
Second, if the Markov chain $p\times p$ is not primitive, the initial move must be chosen so that the limiting distribution is supported on $S\times S$. Third, all actions in $S$ give equal payoff against $p$:
\[
\forall a_1,a_2\in S:\quad \pi^i(\mathrm{All}\,a_1,p)=\pi^i(\mathrm{All}\,a_2,p).
\]
Fourth, no action outside $S$ gives more:
\[
\forall a_1\in S,\ a_2\notin S:\quad \pi^i(\mathrm{All}\,a_1,p)\ge \pi^i(\mathrm{All}\,a_2,p).
\]
Distinct subsets $S$ give disjoint equilibrium classes, and every reactive equilibrium belongs to exactly one minimal $S$ [2606.27653].

The case $S=A$ recovers equalizer strategies. Then the zero restrictions disappear, the outside-action inequalities are vacuous, and only the equal-payoff conditions remain:
\[
\forall a_1,a_2\in A,\quad
g_A^i(a_1)+\sum_{b\in A} g_B^i(b)\,p_{a_1}(b)
=
g_A^i(a_2)+\sum_{b\in A} g_B^i(b)\,p_{a_2}(b).
\]
The paper identifies these with the equalizer or zero-determinant strategies of Press and Dyson and Hilbe et al. [2606.27653].

For a three-action donation game with actions $\{C,M,D\}$ and parameters $b_1>b_2>0$, $c_1>c_2>0$, applying the characterization yields seven equilibrium classes. Evolutionary simulations under Imhof and Nowak dynamics show that when $b_1$ increases from $2.2$ to $4.0$ while $b_2=2$, $c_1=1$, and $c_2=0.5$, $M$-supporting equilibria dominate at low $b_1$, $C$-supporting equilibria dominate at high $b_1$, and the equalizer class $S=A$ is rarely observed because it is far less invasion-resistant [2606.27653]. The maximal number of free parameters in an $m$-action game is
\[
d_{|S|,m}=(|S|-1)^2+(m-|S|)(m-1),
\]
which is symmetric in $|S|$ about $(m+1)/2$ [2606.27653].

## 7. Control sets and equilibrium attainment

A structurally different notion asks which subset of players can be forced to play a target equilibrium so that the remaining players then find that equilibrium a best response. In a finite strategic-form game $G=(N,S,\{u_i\}_{i=1}^n)$ with target equilibrium $\sigma^*\in NE(G)$ and initial profile $s\in S$, a subset $A\subseteq N$ is a direct control set if
\[
\forall i\in N\setminus A:\quad \sigma_i^*\in BR_i\bigl((\sigma_A^*,\, s_{N\setminus A})\bigr).
\]
The minimum size of such a set is denoted $\dcon(G,s,\sigma^*)$ [2312.03229].

This notion links support to equilibrium strength. If every coalition of size at least $m$ is a direct control set for $(s,\sigma^*)$, and if each player weakly prefers $\sigma_i^*$ when the others play $\sigma^*$, namely
\[
\forall s_{-i},\quad u_i(\sigma_i^*,s_{-i})\le u_i(\sigma^*),
\]
then $\sigma^*$ is an $(n-m)$-strong Nash equilibrium, and the bound is tight [2312.03229]. Thus control sets provide a bridge between intervention requirements and robustness against coalitional deviation.

The computational picture is difficult. Deciding whether there exists a direct control set of size at most $k$ is NP-complete, and unless $P=NP$ no polynomial-time algorithm approximates $\dcon(G,s,\sigma^*)$ within factor $n^{1-\epsilon}$ for any $\epsilon>0$, even when each player has only two actions [2312.03229]. Under player-wise monotonicity, however, a Local-Ratio algorithm gives an $f$-approximation when each influence neighborhood has size at most $f$. The paper also identifies tractable or approximately tractable cases inside potential games: singleton congestion games under a general-position assumption and mild cost-monotonicity admit an exact polynomial-time algorithm; symmetric congestion games with decreasing costs can be searched over $k=0,\dots,n-1$ in polynomial time; and in coordination games on graphs, the minimum direct control set problem is NP-complete in the two-colour case, but a target-dominating-set reduction yields an $O(\log \Delta)$-approximation when all $p_i$ are large enough [2312.03229].

In this literature, the “supporting” subset is a subset of players rather than actions or mixed-strategy supports. The conceptual commonality is nevertheless clear: an equilibrium is supported by a small or structured object that certifies its attainability, implementability, or self-enforcement.

Source: https://www.emergentmind.com/topics/s-supporting-equilibria