We seek to understand the effect of adding disruptive highly-capable new technologies to competitions by assessing the addition of Dynamite to Rock-Paper-Scissors. We find that providing a versatile Dynamite move to only one player provides limited value (win probability increases from 50% to 55.5%) and is played rarely. That value decreases further if the game is expanded beyond just the original three moves. We also observe several mechanisms by which prior moves can become strategically unplayable, or obsolete. We hope that this model illustrates some non-intuitive aspects of developing new versatile technologies. We also hope that it illustrates some pitfalls for developers and integrators to avoid in order to create value rather than merely capability.
The Independent Chip Model (ICM) converts tournament chips into reference prize equity, and policies are routinely constructed against those values. Because ICM reads only stack sizes, it omits action order, blind obligations, and seat rotation, and it does not price the elimination pressure a big stack puts on the short stacks it can bust. Those omissions can alter the successor-state contrasts that determine a move. We introduce Strategic-Continuation Optimization (SCO), a policy-construction method that enumerates current-hand outcomes, maps them to successor states, prices those states with continuation values computed from the finite tournament model, and optimizes and freezes the resulting current-hand policy. The fixed-ICM comparison policy changes one thing only: the same optimizer solves the same game with successor states priced by analytic ICM, so the two policies differ only through that pricing. We evaluate the resulting policies in a three-player jam/fold tournament with a \$1M prize pool. Relative to the frozen strategic-continuation benchmark, analytic ICM has \$9{,}036 mean absolute value error across all 2,838 state--seat entries. That value error rewrites the ranges it prices: measured against each decision point's own fixed-ICM jam range, SCO moves the jam frequency by an average of 14.08\%. To price those different moves, we compare all 946 states and three policy owners while changing only the focal policy and holding both opponents and the continuation evaluator fixed. The policy produced by SCO earns \$214.33 more prize equity per hand on average and is favored in 2,433 of 2,838 matched units. The ordering survives replacing the solver-built opponent with two LLMs and with a family of non-modeling threshold players. This value-to-policy-to-cost chain shows directly when ICM becomes an inadequate objective for tournament strategy construction.
A bidder can quietly buy a stake in a company before making an offer for it. That stake, a toehold, is supposed to pay for itself twice: it makes the bidder willing to bid harder, and it frightens rivals into staying out of the fight. The first effect is arithmetic. The second is what would justify the cost and exposure of taking one at all. Yet toeholds are rare in practice, a standing puzzle. We ask whether that second effect is there once the contest is modelled as several rounds of escalating offers rather than the single exchange classical models assume. We turn it into a game a computer can solve, and certify the answers to an accuracy a referee can check. Three findings. The auction fixes what the toehold-holder earns but not how it bids: the same contest supports a bidder who opens aggressively against a rival who folds, and one who opens cheaply against a rival who does not, with the same profit either way. Aggressive preemptive bidding still appears when the toehold is removed entirely, so it comes from bidding in public and in turns, not from owning the stake. And the tidy "bigger toehold, more deterrence" relationship holds only in a contest cut short after one round; give it a real second round and it stops responding. So the two reasons to buy a toehold do not fare alike. The profit reason holds up; the deterrence reason does not, which suggests why toeholds may be rarer than theory predicts, alongside the procedural costs of disclosure and price impact that this model omits. A warning follows for anyone computing economics from a game solver: solve this auction once and it returns a confident figure for what a preemptive bid is worth; solve it again from a different start and it returns a different one, equally converged. We also report which solvers cope with contests of this shape, including versions too large to enumerate. Code is released.
Organizations often pool dispersed information into one ranking and then allow many agents to act on that shared view. In a discovery problem, this can improve beliefs while reducing coverage. We develop an exactly solvable benchmark with sixteen boxes, one target, eight searchers, and noisy private clues. Pooling raises the accuracy of the best single recommendation from 0.20 to 0.3835, but repeating that recommendation lowers group discovery from 0.8322 under decentralized clue-following to 0.3835. A coordinated eight-action portfolio using the same pooled reports reaches 0.8594, and seven coordinated actions recover the decentralized benchmark. The paradox is a protocol failure, not an information failure: a one-answer rule compresses a portfolio of available actions into one repeated choice. We then replace the planner with self-interested searchers who split a prize. The equal-split game is a potential game. Its anonymous symmetric equilibrium obeys a water-filling rule. In the canonical instance it achieves 0.5991: strictly above consensus, but below both private search and the planner. The exact mixed price of anarchy is 2 - 1/N. A sole-rescue reward, which pays only an agent who covers the target alone, makes every pure Nash equilibrium first-best. Finally, a latent common-cue model shows how correlated reports collapse effective discovery channels. The centralized planner gain rises strictly with copying, and in the canonical environment the symmetric market overtakes decentralized report-following at copying probability c = 0.788462. In a proportional large-market limit the five-protocol ordering survives exactly: consensus discovery vanishes while blind, market, private, and portfolio search converge to 0.500, 0.547, 0.847, and 0.874. The contribution is a compact benchmark that separates information, allocation, incentives, and dependence into exact, reusable quantities.
This paper develops a multiscale model of coalition formation in which strategic exit-and-join decisions are coupled with tactical consensus dynamics inside coalitions. Coalition value is generated endogenously from within-coalition information aggregation, while Aumann-Dreze payoffs, switching frictions, and acceptance rules govern strategic reconfiguration. The framework introduces a fast-slow architecture in which transferable coalition value emerges from DeGroot-style consensus processes, while coalition structures evolve through incentive-driven exit-and-join dynamics. The analysis characterizes joint tactical-strategic equilibria, conditions for tactical and strategic unanimity, segregation, polarization, and cognitive barriers that sustain stable coalition structures. A fixed-point characterization and existence results are established for the coupled dynamics. Numerical experiments reveal an instability-consensus paradox: low or negative switching barriers may prevent strategic convergence while simultaneously promoting temporal mixing sufficient to achieve global tactical consensus. The results provide a unified perspective on coalition formation, consensus dynamics, information aggregation, and strategic stability in multi-agent systems.
This paper studies coalition formation as a decentralized dynamical process driven by unilateral exit-and-join decisions. Agents evaluate local moves using the Aumann-Dreze value, so payoffs are computed within the agent's current coalition rather than through a globally negotiated coalition structure. The resulting model links cooperative payoff allocation with noncooperative best-response behavior: a terminal partition is precisely a coalition structure with no admissible, individually profitable exit-and-join deviation. We establish equilibrium characterizations, identify conditions under which the dynamics admit scalar Lyapunov or exact-potential representations, and analyze how switching and acceptance costs shape local stability. Numerical experiments test finite-time stabilization, cost sensitivity, and a special convex-game benchmark.
We provide a strategic foundation for information: in any given game with incomplete information we define strategic quotients as information representations that are sufficient for players to compute best-responses to other players. We prove 1/ existence and essential uniqueness of a minimal strategic quotient called the Strategic Type Space (STS) in which a type is given by an interim correlated rationalizability hierarchy and represents a set of beliefs over other players' types and nature that rationalize this hierarchy and 2/ that the minimal STS has a recursive structure that is captured by a finite automaton.