What library of biological dynamic games comes with a companion Julia solver?

Twenty-five games from molecules to populations

CorleoneGame is a library of 25 biological dynamic games, each with 2 to 10 players, covering molecular, cellular, organismal and population scales. The interactions span competitive, antagonistic, mutualistic and regulatory types. Instances describe single seasons, cycles that return to their initial state, or episodic tasks; seven include duration choice, and in some games the players' feasible sets are coupled rather than independent.

Each game ships with four things: a biological motivation, an executable Julia specification that assigns controls, objectives and restrictions to named players, a parameter instance, and a reference open-loop generalized Nash equilibrium candidate with interpreted trajectories. The abstract frames the underlying idea plainly: a dynamic game makes an interaction explicit by stating who controls which decision, what each participant values, and which restrictions constrain unilateral deviations.

Sequential best responses, then an audit

CorleoneGame.jl does not solve for an equilibrium in one shot. It computes candidates by sequential optimal-control best responses, then audits each candidate for feasibility and for profitable unilateral deviations on a coarse and a refined control grid. The audit samples path restrictions at eight equidistant points per control interval, so it is at least as dense as the optimization grid.

The audit result is the paper's headline quantitative claim: all 25 reference computations satisfy the audit tolerances on both grids. Reintegrating every saved final control profile with the released model definitions reproduces the saved terminal states to double precision, with a maximum sampled violation of 9.1·10⁻⁷, below the acceptance tolerance ε_g = 10⁻⁶. All final deviation-audit calls on both grids report Success for every instance; in one instance, motor, nine of 140 intermediate and final response calls report MaxIters, while the other 131 report Success.

An audited candidate is not a certified equilibrium

The reference candidates are open-loop, need not be unique, and are accepted on two control grids by local solves. They carry no certificate of an exact continuous-time equilibrium. The sequential iteration has no general convergence guarantee, and in some periodic instances only a looser return band yields convergence. Most parameters are synthetic.

These are properties of the method, not incidental caveats. A reader comparing candidates across games should treat them as audited reference points on a stated grid, not as unique solutions. The paper also notes that generalized Nash equilibria are typically not isolated, which matters for the inverse-game use described below: a recovered specification can reproduce data through another equilibrium of the same game, and a correct specification can fail to reproduce them through equilibrium selection.

The plant–herbivore game reproduces a prediction only qualitatively

The running example is tupelo, a plant–herbivore game mapped from a published scenario. Its executable specification yields an audited reference candidate that qualitatively reproduces the published delayed mass hatching, but not the complete conflict avoidance. The paper attributes that gap to the open-loop information structure: in the source model the plant can respond to current larval density with induced defense, so the threat of defense alone deters early hatching, whereas a schedule fixed at the start of the season cannot respond to larval density, and defense must therefore be used rather than only threatened.

This comparison is the clearest place where the library's design choice has a biological consequence. The same specification machinery that makes preferences and restrictions explicit also fixes when information is available, and that timing changes which behavioral outcome the candidate can express.

Biological hypotheses become comparable instance variants

Because all instances share one specification schema, alternative biological hypotheses become instance variants that can be solved and compared with the same method. The tupelo example shows the chain: an executable specification yields an audited reference candidate; changes of preferences, restrictions or mechanisms turn hypotheses into variants; and the candidates generate synthetic observations whose hypothesis space for inverse games is defined by those variants.

The paper is explicit that no inverse experiment is reported here. It also lists Level-4 difficulties the library allows to study in a controlled way: permutation symmetry of players with similar roles, cooperative mimicry of a single aggregate optimizer, and an unknown number of players, which the variable-N instances can vary. Reports using this route should state the forward solver, its initializations, and whether several observation scenarios or initializations were used to assess ambiguities between weights, restrictions and equilibrium selection.

One candidate shows how weights shape trajectories

The plantNitrogen instance makes the weight-to-trajectory link concrete. Three plants compete for an early-spring nitrogen pulse; all start at maximal uptake and draw soil nitrogen from 2 down to the uptake threshold of 0.2 within less than two time units, after which the shared floor limits joint uptake. Plant A, which weighs effort costs most, drops to about 0.3 almost immediately; B keeps an intermediate schedule; C, which values carbon most, maintains high uptake longest and exhausts its effort budget. Plant nitrogen peaks earliest and most moderately for A (t≈1.2), later for B (t≈3), and latest and highest for C (t≈8).

The interpretation is the authors': the differing weights, not a fitted parameter, drive the divergence in schedules. A reader using this instance should note that all rates, budgets, weights and initial values are synthetic, dimensionless benchmark choices rather than fitted values.

Where the library transfers and where it stops

The library transfers as a specification and benchmarking resource: 25 executable games, a common schema, saved reference candidates, and scripts that generate all tables and figures. The paper states that no experimental data were used and that the saved candidates and synthetic observations are part of the code repository. Each instance is also documented in the dynamic-games category of the benchmark library mintoc.de, so new instances can be added under the same classification system.

It stops short of calibrated biology. Most parameters are synthetic, the candidates are open-loop and non-unique, and the sequential iteration has no general convergence guarantee. The paper's own next steps point to inverse games starting from the hypothesis spaces, ideally with real-world data; feedback equilibria for selected instances to test whether conclusions such as the tupelo comparison depend on the information structure; and calibrated instances, multiseason models with an explicit offseason map, and scalability studies with a variable number of players.

Check the information structure before comparing candidates

The most decision-relevant distinction in this paper is not the count of 25 games but the information structure behind each candidate. The tupelo comparison shows that an open-loop schedule can fail to reproduce a prediction that a feedback-capable model achieves, because the plant cannot respond to current larval density. Before treating any reference candidate as evidence about a biological hypothesis, a reader should check whether the hypothesis depends on responding to a state that the open-loop schedule cannot see.

The second check is the audit itself. All 25 reference computations pass on both grids, but that pass is a statement about feasibility and profitable unilateral deviations on those grids, not a certificate of an exact continuous-time equilibrium. The paper's own priority ordering follows from this: feedback equilibria for selected instances come first, because they determine which hypotheses remain plausible for given synthetic or real-world data.

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