Randomized Local Coordination: Mechanisms & Applications

• Randomized local coordination is a decentralized paradigm where agents use only local information and stochastic rules to achieve global collective behavior.
• It leverages mechanisms such as replicator dynamics, best response, and random sampling to drive efficient equilibrium selection and cost reduction.
• Applications span distributed optimization, multi-agent decision-making, and load balancing, emphasizing trade-offs between connectivity, topology, and randomness.

Searching arXiv for the cited work and closely related papers on local coordination, decentralized optimization, distributed coordination, and local distributed decision. Randomized local coordination denotes a class of decentralized processes in which agents act using only local information, local interaction neighborhoods, or limited-radius communication, while stochasticity enters through random interaction structure, random update rules, random sampling, or shared and private randomness. Across game theory, decentralized optimization, distributed algorithms, and information theory, the term encompasses mechanisms in which global collective behavior emerges from local randomized decisions rather than centralized control. In the most direct game-theoretic formulation, agents occupy nodes of a random network and repeatedly update strategies based on neighboring states; in optimization, nodes coordinate only on randomly selected local regularizers; in distributed computing, nodes solve locally checkable tasks with local randomness; and in strong coordination, terminals communicate over a sparse network while using randomness to synthesize a prescribed joint action distribution (Raducha et al., 2021, Lin et al., 17 Sep 2025, Vellambi et al., 2016).

1. Conceptual scope and formal settings

The broadest common structure is a networked population of agents, processors, or nodes; a local information constraint; and a randomized mechanism that substitutes for global synchronization. Different literatures instantiate these ingredients differently. In coordination games on graphs, each player interacts only with graph neighbors and updates asynchronously according to a local rule such as replicator dynamics, best response, or unconditional imitation (Raducha et al., 2021). In decentralized optimization with partially separable regularization, each node samples one regularizer component uniformly and coordinates only with nodes participating in that term, thereby replacing a global proximal step by a randomized local one (Lin et al., 17 Sep 2025). In strong coordination over line networks, agents communicate over rate-limited links and use common randomness together with local randomness to generate action sequences whose joint law approximates a target distribution in total variation (Vellambi et al., 2016).

A unifying technical distinction is between randomized local coordination as a means of selecting among multiple equilibria, and randomized local coordination as a means of reducing communication or computational cost. The first appears in networked coordination games, where randomness is in graph structure or behavioral noise and the principal questions are whether agents coordinate at all and which equilibrium is selected (Raducha et al., 2021, Jones et al., 2021). The second appears in optimization and distributed learning, where randomness is used to sample coordinates, edges, or regularizers so that only a small subset of agents coordinate per iteration while preserving convergence guarantees (Lin et al., 17 Sep 2025, Ying et al., 2018, Liu et al., 14 Apr 2026).

The notion also has a complexity-theoretic interpretation in distributed computing. In locally checkable labeling problems, nodes must output labels based on bounded-radius neighborhoods, and the central question is how much randomization improves local coordination relative to deterministic protocols (Balliu et al., 2019, Fraigniaud et al., 2010). On trees, many classes of such problems exhibit no asymptotic advantage from stronger randomized local models over deterministic LOCAL, indicating that graph geometry can fundamentally constrain local coordination even when randomization is abundant (Dhar et al., 2024).

2. Game-theoretic foundations on networks

A canonical model is the study of two-player coordination games played on random graphs, where agents occupy nodes of random regular or Erdős–Rényi networks and update asynchronously via local evolutionary rules (Raducha et al., 2021). Two classes of games are central. The pure coordination game with two equivalent strategies has payoff matrix

$\begin{array}{c|cc} & A & B \ \hline A & 1 & 0 \ B & 0 & 1 \end{array}$

and asks whether the system reaches full coordination or freezes in a disordered coexistence state. The more general coordination game is normalized as

$\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$

with $S<0$ and $T<1$, so that coordination on $A$ is payoff-dominant while risk dominance is determined by the Harsanyi–Selten boundary $T=S+1$ (Raducha et al., 2021).

Three update rules sharply separate the role of locality. Under replicator dynamics, a node compares payoff with a randomly chosen neighbor and copies the neighbor’s strategy with probability proportional to the payoff difference. Under myopic best response, a node computes which action yields the higher payoff against the current neighborhood configuration and switches accordingly. Under unconditional imitation, a node copies the highest-payoff neighbor if that payoff exceeds its own (Raducha et al., 2021). Mean-field analysis and simulations agree that replicator dynamics and best response largely preserve classical risk-dominance predictions on random graphs, whereas unconditional imitation is substantially more sensitive to graph sparsity and finite-size effects.

For the pure coordination game, all three rules exhibit a transition from disordered frozen states to full coordination as connectivity increases. For replicator dynamics and best response on random regular graphs, the critical connectivity is

$k_c^{RD}=4,\qquad k_c^{BR}=4,$

and this threshold is system-size independent across the reported range $N=10^3$ to $10^4$ (Raducha et al., 2021). For unconditional imitation, the transition is markedly different: at $N=1000$, the critical connectivity is approximately $\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$0, full coordination typically requires much higher degree, and finite-size scaling is consistent with

$\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$1

This shows that global coordination remains achievable below complete-graph density, but only through a connectivity threshold that itself grows with system size (Raducha et al., 2021).

In general coordination games, the most salient finding is equilibrium selection. For replicator dynamics and best response, the phase boundary remains the risk-dominance line $\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$2, independent of degree and system size, so randomized local interaction structure does not overturn mean-field equilibrium selection (Raducha et al., 2021). Under unconditional imitation, however, sparse random graphs can select the payoff-dominant equilibrium even when it is not risk-dominant. This effect disappears on complete graphs, where unconditional imitation becomes effectively equivalent to best response. In the special case $\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$3, the equilibrium-selection threshold satisfies

$\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$4

for moderate $\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$5, approaching the risk-dominance threshold $\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$6 only as $\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$7 (Raducha et al., 2021). The combination of local clustering and winner-take-all imitation therefore creates a regime in which decentralized local interaction favors efficient coordination over safer but inefficient equilibria.

A related but distinct form of randomized local coordination appears in network coloring problems. On random bipartite graphs guaranteed to be 2-colorable, myopic artificial agents update using local conflict information and occasional random moves (Jones et al., 2021). The basic greedy rule can become trapped in local minima, exemplified by a six-node bowtie motif whose gridlock probability under pure greedy dynamics is

$\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$8

Randomness-first and memory-based rules inject stochastic exploration at different stages of the local decision process. The paper shows that both the fraction of noisy agents and the location of the noise in the update rule matter: memory-1, which randomizes only at conflict nodes in stagnant neighborhoods, can drive the probability of unsolved networks near zero across tested sizes and degrees, whereas too much unguided noise degrades performance (Jones et al., 2021). This establishes a second game-like mechanism: occasional local random exploration can destabilize absorbing suboptimal states and enable eventual global coordination.

3. Geometry, amenability, and limits of local coordination

The feasibility of local coordination depends not only on update rules but also on network geometry. In a pure coordination game on a large social network, where agents choose actions in $\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$9 and communicate only within graph distance $S<0$0, a central quantity is inefficiency

$S<0$1

the normalized fraction of mismatched edges (Hutchcroft et al., 13 Feb 2026). A finite graph is $S<0$2-amenable if one can partition vertices into connected components of radius at most $S<0$3 after removing at most $S<0$4 edges. This notion captures when near-efficient local coordination is structurally possible.

The principal result is a near-equivalence between local coordination and amenability. If a graph is $S<0$5-amenable, then leader-based local communication protocols can coordinate almost perfectly within each community, with disagreements only on the removed boundary edges, so inefficiency is at most $S<0$6 or a slightly worse constant-factor variant depending on whether private messages are available (Hutchcroft et al., 13 Feb 2026). Conversely, if there exists any action-symmetric radius-$S<0$7 strategy profile with inefficiency at most $S<0$8, then the graph must be $S<0$9-amenable (Hutchcroft et al., 13 Feb 2026). Thus non-amenable graphs such as expanders or tree-like random networks cannot support arbitrarily good randomized local coordination.

This converse was sharpened in the binary unbiased setting using Shapley values of a mutual-information game. If $T<1$0 are unbiased local outputs measurable with respect to independent input variables in $T<1$1, and if the average disagreement obeys

$T<1$2

then the graph is $T<1$3-amenable, where $T<1$4 is the binary entropy function (Peretz et al., 1 Jun 2026). Since $T<1$5 for small $T<1$6, this improves the previous square-root converse to an $T<1$7 amenability bound in the binary case (Peretz et al., 1 Jun 2026). A plausible implication is that unbiased local randomized coordination is more tightly constrained by graph expansion than variance-based analyses alone suggest.

The same geometric viewpoint yields optimality statements on specific graph families. On large cycles, the best achievable inefficiency under $T<1$8-local correlated equilibrium scales as $T<1$9, and leader equilibria attain exactly this rate, so local randomized coordination is essentially optimal there (Hutchcroft et al., 13 Feb 2026). By contrast, on non-amenable graphs no local protocol can produce high pairwise correlations over edges. This provides a structural answer to a common misconception: failures of local coordination on sparse graphs are not always due to poor algorithm design; in many cases they are imposed by the network’s coarse geometry.

4. Decentralized optimization and learning with randomized local coordination

In decentralized optimization, randomized local coordination is used to replace full synchronization by sparse, stochastic interaction. A representative problem is

$A$0

with

$A$1

where each $A$2 depends only on a subset $A$3 of nodes (Lin et al., 17 Sep 2025). In graph-guided regularization, each $A$4 is pairwise with $A$5, corresponding to an edge regularizer. Standard proximal-gradient methods require evaluating $A$6, which induces $A$7 communications per iteration and hence scales with topology size.

The BlockProx algorithm replaces the proximal map of the full sum by the proximal map of a single randomly sampled regularizer. At iteration $A$8, each node computes a local gradient step

$A$9

samples an index $T=S+1$0, and updates

$T=S+1$1

with $T=S+1$2 (Lin et al., 17 Sep 2025). The crucial communication result is that if $T=S+1$3, then the expected number of messages per iteration is

$T=S+1$4

For graph-guided regularizers, $T=S+1$5 for all $T=S+1$6, so the expected communication cost is exactly 2 messages per iteration, independent of the number of nodes, the number of edges, and graph topology (Lin et al., 17 Sep 2025).

Despite this drastic communication reduction, convergence is preserved. Under convexity and bounded subgradients or smoothness, the method achieves $T=S+1$7 iterations to $T=S+1$8-suboptimality; under strong convexity, it achieves $T=S+1$9 to an $k_c^{RD}=4,\qquad k_c^{BR}=4,$0-solution with decaying stepsizes or $k_c^{RD}=4,\qquad k_c^{BR}=4,$1 to a neighborhood with constant stepsize (Lin et al., 17 Sep 2025). The analysis hinges on an unbiased expected-distance recursion over randomly selected components and proximal nonexpansiveness. Experiments on synthetic multitask regression and a Sacramento housing dataset show substantially better accuracy than ADMM, proximal averaging, distributed subgradient descent, and Walkman under fixed communication budgets, with empirical communication per iteration remaining very close to 2 (Lin et al., 17 Sep 2025).

A related but older instance appears in dynamic average diffusion with randomized coordinate updates. Each agent in a network observes a time-varying vector $k_c^{RD}=4,\qquad k_c^{BR}=4,$2 and seeks to track the network average

$k_c^{RD}=4,\qquad k_c^{BR}=4,$3

Instead of exchanging full vectors, agents update only one randomly selected coordinate at a time. In the synchronized-coordinate case, all agents select the same coordinate $k_c^{RD}=4,\qquad k_c^{BR}=4,$4; in the independent case, each agent selects its own coordinate $k_c^{RD}=4,\qquad k_c^{BR}=4,$5 independently (Ying et al., 2018). Because independent coordinate choices destroy the doubly stochastic mixing structure, naive schemes become biased. The paper resolves this by introducing push-sum normalization together with memory variables $k_c^{RD}=4,\qquad k_c^{BR}=4,$6 that store the most recently observed values of each coordinate (Ying et al., 2018). The result is unbiased tracking with mean-square convergence, even under independent coordinate choices, while reducing communication by a factor of $k_c^{RD}=4,\qquad k_c^{BR}=4,$7.

Open multi-agent systems provide yet another optimization-centric example. In optimal resource allocation with constraint $k_c^{RD}=4,\qquad k_c^{BR}=4,$8, random coordinate descent selects a random pair of agents $k_c^{RD}=4,\qquad k_c^{BR}=4,$9 and updates only those two coordinates while preserving feasibility (Galland et al., 2021). On complete graphs with homogeneous agents and strongly convex smooth local objectives, the closed-system iteration obeys

$N=10^3$0

where $N=10^3$1 is the condition number (Galland et al., 2021). When agents are occasionally replaced, the error follows a linear recursion with additive disturbance; stability requires the replacement-to-update ratio to satisfy

$N=10^3$2

This quantifies how much openness randomized pairwise local coordination can absorb before losing track of the moving minimizer (Galland et al., 2021).

5. Distributed computing and local decision

In distributed graph algorithms, randomized local coordination is studied through locally checkable labeling problems, where global feasibility reduces to local consistency constraints. Classic examples include sinkless orientation, maximal independent set, and graph coloring (Balliu et al., 2019). It was long known that some such problems benefit exponentially from randomization—for example, sinkless orientation requires $N=10^3$3 rounds deterministically in the LOCAL model but only $N=10^3$4 rounds randomized. A key result is that this dichotomy is not exhaustive: there exist LCLs with deterministic complexity $N=10^3$5 and randomized complexity $N=10^3$6, obtained via a padding construction that inflates the communication radius needed to simulate a base problem (Balliu et al., 2019). More generally, the paper constructs a hierarchy of LCLs with deterministic complexity $N=10^3$7 and randomized complexity $N=10^3$8 for any constant $N=10^3$9 (Balliu et al., 2019).

This line of work demonstrates that randomization can accelerate local coordination without changing the local nature of verification. Nodes still act based on bounded-radius neighborhoods; random bits simply reduce the amount of deterministic symmetry breaking or neighborhood exploration needed. A plausible implication is that randomized local coordination can be understood as a resource trading local certainty for fewer communication rounds, but only within complexity bands permitted by the problem’s combinatorial structure.

The role of randomness in local decision is subtler. In the LOCAL model, a distributed language belongs to LD if it can be decided in a constant number of rounds deterministically, and to BPLD if it can be decided with bounded error in constant rounds using randomization (Fraigniaud et al., 2010). For general languages, randomization can strictly help: the uniqueness-of-leader language is not in LD for any $10^4$0 rounds but belongs to $10^4$1 whenever $10^4$2 (Fraigniaud et al., 2010). For hereditary languages, however, randomization does not help once correctness parameters are sufficiently strong: if $10^4$3 with $10^4$4, then $10^4$5 (Fraigniaud et al., 2010). Thus local randomness can coordinate distributed decisions only under specific structural and error-tolerance regimes.

Recent results on trees extend this structural theme. The randomized online-LOCAL model subsumes classical LOCAL, quantum LOCAL, finitely dependent distributions, SLOCAL, dynamic-LOCAL, and other causality-respecting models. Yet for many LCL classes on trees, randomized online-LOCAL offers no asymptotic improvement over deterministic LOCAL (Dhar et al., 2024). On rooted regular trees, solvable LCLs fall into four classes: $10^4$6 in both models, $10^4$7 in deterministic LOCAL but $10^4$8 in randomized online-LOCAL, $10^4$9 in both, and $N=1000$0 in both for finite depth $N=1000$1 (Dhar et al., 2024). On unrooted regular trees, super-logarithmic localities also coincide (Dhar et al., 2024). This shows that graph structure can erase the benefit of stronger randomized local coordination models, including quantum variants.

6. Information-theoretic strong coordination and related mechanisms

An information-theoretic extreme of randomized local coordination is strong coordination over communication networks. In a three-terminal line network, Agent 1 communicates to Agent 2 at rate $N=1000$2, Agent 2 communicates to Agent 3 at rate $N=1000$3, and all agents share common randomness of rate $N=1000$4 (Bloch et al., 2013). The goal is to ensure that action sequences $N=1000$5 are close in total variation to an i.i.d. target law $N=1000$6. The coordination capacity region admits inner and outer bounds expressed via auxiliary variables $N=1000$7, with a rate region

$N=1000$8

defined by seven mutual-information inequalities, including

$N=1000$9

together with sum-rate constraints over $\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$00 (Bloch et al., 2013). When common randomness is abundant, the projected region simplifies to

$\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$01

showing explicitly that common randomness reduces communication costs relative to explicit action transmission (Bloch et al., 2013).

The multi-hop generalization introduces $\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$02 nodes on a line, local randomness $\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$03 at each node, common randomness $\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$04, and hop-by-hop rates $\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$05. A strong coordination code must satisfy

$\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$06

The paper derives general inner bounds via multilayer channel resolvability codes built from auxiliary variables $\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$07, and proves optimality in several special regimes, notably functional mode and sufficiently large common randomness (Vellambi et al., 2016). In the latter regime, local randomness becomes dispensable and link rates collapse to

$\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$08

This formalizes a deep version of randomized local coordination: agents at each hop use local and shared random seeds plus messages to synthesize a globally specified stochastic process (Vellambi et al., 2016).

Another resource-allocation example is randomized constraints consensus for distributed robust linear programming. Each processor knows only its own uncertain constraints and a common objective. At each round, it performs randomized local verification by sampling uncertainty realizations and checking feasibility of its current candidate solution, then exchanges only its active basis with neighbors and resolves a local LP using its own basis, neighbors’ bases, and any sampled violating constraint (Chamanbaz et al., 2017). If the verification step is successful for sufficiently many rounds, consensus has been reached; the resulting solution is feasible and optimal with high confidence up to a violation set of arbitrarily small probability measure (Chamanbaz et al., 2017). This mechanism combines local randomness in constraint sampling with sparse local communication over time-varying directed graphs.

7. Randomized local coordination in scheduling and load balancing

The term also appears in coordination mechanisms for selfish scheduling. In unrelated-machine scheduling with objective $\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$09, local policies specify how jobs assigned to the same machine are ordered. Deterministic strongly local non-preemptive policies, exemplified by Smith’s rule, have price of anarchy 4, and this is optimal within that class (Cole et al., 2010). Randomization improves coordination. The randomized non-preemptive policy Rand orders jobs on machine $\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$10 so that for any pair $\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$11,

$\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$12

and achieves price of anarchy at most $\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$13, improving to $\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$14 when total weighted processing time is negligible relative to optimal social cost (Cole et al., 2010). The policy induces an exact potential game, so pure Nash equilibria always exist (Cole et al., 2010). Here randomization symmetrizes externalities and smooths worst-case interactions that deterministic local ordering cannot avoid.

A different setting is balls-into-bins load balancing via randomized local search. Each ball owns an exponential clock; when activated, it samples a random bin and moves if that bin has smaller load than its current bin (Berenbrink et al., 2017). This entirely local, selfish rule drives the system toward perfect balance. The expected time to reach perfect balance from an arbitrary initial configuration is

$\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$15

which is asymptotically tight (Berenbrink et al., 2017). The key analytical device is the destructive-moves lemma: adding arbitrary reversals of beneficial moves cannot accelerate balancing, which permits worst-case simplifications and phase-based analysis (Berenbrink et al., 2017). This is randomized local coordination in perhaps its purest operational form: each agent knows only current and sampled destination loads, yet the collective system converges to a globally balanced state at an optimal timescale.

8. Synthesis, misconceptions, and research directions

Across these literatures, three recurrent determinants govern the success of randomized local coordination: update-rule structure, topology or geometry, and the form of randomization. Update rules matter because local randomness can either destabilize local minima, as in memory-based coloring rules (Jones et al., 2021), or amplify local efficient clusters, as in unconditional imitation on sparse coordination graphs (Raducha et al., 2021). Topology matters because sparse expanders, trees, or non-amenable graphs impose hard limits on what local rules can correlate, whereas amenable or hyperfinite graphs admit near-efficient decomposition-based coordination (Hutchcroft et al., 13 Feb 2026, Peretz et al., 1 Jun 2026, Dhar et al., 2024). The form of randomization matters because unbiased stochastic approximation can preserve convergence in optimization (Lin et al., 17 Sep 2025), while naive weighted Markovian sampling can cause entrapment unless exploration is restored through jumps (Liu et al., 14 Apr 2026).

A frequent misconception is that more connectivity always improves coordination. In several settings the opposite occurs. In networked coordination games, unconditional imitation selects the payoff-dominant equilibrium only below a critical connectivity and loses this property on dense graphs (Raducha et al., 2021). In decentralized learning, weighted Metropolis–Hastings random walks can become trapped in highly connected but locally biased regions, causing slower convergence than simpler sampling schemes (Liu et al., 14 Apr 2026). Another misconception is that randomization automatically compensates for poor topology. The amenability converses and tree-model classification show that some structural barriers are not algorithmic but geometric: no amount of local randomness, or even stronger causal or quantum models, removes them in broad problem classes (Hutchcroft et al., 13 Feb 2026, Dhar et al., 2024).

Several research directions remain open. In optimization, constants and limiting neighborhoods often still scale with the number of regularizer components $\begin{array}{c|cc} & A & B \ \hline A & 1 & S \ B & T & 0 \end{array}$16, suggesting a need for structure-aware sampling beyond uniform term selection (Lin et al., 17 Sep 2025). In open multi-agent systems, extending random coordinate descent guarantees from complete graphs and homogeneous agents to sparse, heterogeneous, or asynchronous settings remains unresolved (Galland et al., 2021). In token-based decentralized learning, the trade-off between improved spectral gap and stationary-distribution bias under jump kernels invites adaptive control of jump probability and nonreversible transition design (Liu et al., 14 Apr 2026). In distributed computing, the full landscape of randomized versus deterministic locality gaps beyond current gadget constructions is still not known (Balliu et al., 2019). In strong coordination, the exact unrestricted multi-hop rate region for general target distributions remains open (Vellambi et al., 2016).

Taken together, randomized local coordination is not a single algorithmic template but a cross-disciplinary principle: replace global synchronization by bounded-radius or sparse local interaction, inject randomness at the level of agents, links, regularizers, or messages, and exploit stochastic averaging or clustering to recover desirable global behavior. Its success depends delicately on whether the problem’s objective, the update law, and the network’s structural constraints are aligned. When they are, local randomness can reduce communication, improve equilibrium selection, accelerate exploration, and realize complex joint behavior. When they are not, topology and locality impose sharp, sometimes information-theoretic limits (Raducha et al., 2021, Lin et al., 17 Sep 2025, Hutchcroft et al., 13 Feb 2026).