*First paper in Footnotes on the Foundations of Game Theory. It provides verifiable information of a universal solver g[], for finite pure-strategy Nash Equilibria, without disclosing code. In doing so, theory at work is displayed. In particular, this feat exploits a bridge between static games and Replicator dynamics to show that only predicted symmetric NE are stable fixed points. A theoretical discussion from the perspective of a Machiavellian ruler accompanies this mathematical treatment.*

The second part establishes a link between the number of solutions and their character in terms of symmetric (diagonal) solutions and coupled (off-diagonal) in symmetric matrices with heuristics. Data generated with **g[] **is used to test hypotheses. This method is improved on with a statistical model to predict the distribution of the number of solutions. Data supports the model. All results follow from simple or common-knowledge concepts and definitions. More general solvers will be discussed in future work.

**Keywords:**

*Game Theory*, *Applied Mathematics*, *Social Science*, *Evolution*.

**Please note:*** This paper is a first draft, revision may result in major updates*. Comments, requests or questions are welcome.

You may contact me at manuel.echeverria.q@gmail.com

# UNIVERSAL SOLVERS [1/2]

This text belongs in the series *Footnotes on the Foundations of Game Theory*. It introduces the universal solver **g[] **for pure-strategy Nash Equilibria in static finite games[1]. This presentation establishes the feasibility of the project, and provides glimpses of its scientific potential, beyond the ability of solving thousands of games instantly. In particular, it is feasible to make and test predictions about a large number of games or huge ones. Ultimately, this paper embarks on a route to explore the set of all possible games, in turn the basis for a wide range of thought experiments and theorems within the Sciences, Mathematics and Philosophy. Theory is discussed alongside the solver, from the perspective of a Machiavellian ruler.

It is known mathematicians are able to provide proofs without revealing much about of the underpinnings of their work. Likewise, this paper does not print code but provides information to check the following constraints:

- Proposed solutions indeed are NE.
- All other outcomes are unstable.
- Games are not rigged in order to falsely emulate a solver.
- Provide data to check all of the above.

The reader will also be introduced to fundamental connections between seemingly disparate fields. In particular, this paper explores the connection between static and dynamic games, with implications on a much heated debate. Satisfying a-d amounts to problem solving, which displays theory at work in the process.

This Footnote starts by generating of data from known distributions, and then proceeds with automated solving by virtue of **g[]**. The cost of deception (c), is high and increasing because manipulations nevertheless must (i) be such that all *pure-strategy* NE are found (ii) conform to a well-known distribution with specified parameters. In addition (iii), big enough games will be computed in sizeable quantities. This combination will work as a deterrent burdening deception in proportion to the number of games solved with transparency. Future *Footnotes on the Foundations of Game Theory* will expand on universal mixed-strategy solvers, dynamic games, Markov processes and Evolutionary Game Theory. In addition, predicting NE outcomes from a particular distribution of the incentive structure has theoretical appeal. This paper takes important steps in that direction, and will be expanded on in updates and future footnotes.

**To check large games** is cumbersome and arguably requires a similar device in the hands of the readers. In order to address this while respecting a-d, Evolutionary Game Theory will be used to provide means to evaluate results. Articulation of games in terms of *Replicator Dynamics* bridges symmetric NE and fixed points, where myopic players adapt to the average (field) play. In this manner I effectively provide data and well-known equations to check the solution, without disclosing crucial information about **g[]** etc. I will demand some goodwill and/or effort from the readers when evaluating a batch of 50, seven-player randomly generated games and their solutions. The sheer size should be enough to deter manipulation and/or manual search. The Replicator Dynamics approach is introduced with a (50×50) 2-player game in order to avoid more theorems or analysis resulting from n>2 *players*. In this setting, the *strategies* can be interpreted as *types* of a *population*.

**Elusive topics** such as the stark connection between method, theory and ruminations on society are given systematic treatment. In similar fashion, informal reasoning on the correspondence between evolution, myopic behaviour and rational behaviour in society is given sound theoretical basis.

# DATA g [ ]

50 games with seven players are solved. The number of strategies for each player is drawn from a *Uniform Distribution*[1,3]. The incentive structure is generated with a *Poisson* of mean 3 for each player strategy; and at each contingency of the game. All relevant data is uploaded with description.

# RESULTS 1

Strategy indexed are reserved for each player with indexes: ABC, DEF,… STU. Table 1 provides all the states & NE for the first of the 50 games.

**T1. NE Game 1/50**

STATE | P1 | P2 | P3 | P4 | P5 | P6 | P7 | STATE | P1 | P2 | P3 | P4 | P5 | P6 | P7 |

ADGJMPS | 4 | 1 | 2 | 5 | 5 | 2 | 4 | CDGJOPS | 2 | 2 | 3 | 3 | 2 | 1 | 2 |

ADGJNPS | 5 | 3 | 1 | 4 | 9 | 4 | 7 | CDGKMPS | 5 | 2 | 4 | 3 | 1 | 2 | 2 |

ADGJOPS | 2 | 2 | 4 | 4 | 2 | 4 | 6 | CDGKNPS | 3 | 3 | 5 | 2 | 3 | 4 | 1 |

ADGKMPS | 6 | 2 | 2 | 4 | 2 | 3 | 0 | CDGKOPS | 5 | 4 | 5 | 5 | 7 | 1 | 1 |

ADGKNPS | 6 | 3 | 3 | 3 | 2 | 2 | 6 | CDGLMPS | 4 | 4 | 4 | 3 | 7 | 2 | 2 |

ADGKOPS | 3 | 2 | 2 | 2 | 6 | 4 | 1 | CDGLNPS | 2 | 3 | 2 | 2 | 0 | 5 | 2 |

ADGLMPS | 3 | 1 | 2 | 1 | 2 | 5 | 3 | CDGLOPS | 3 | 3 | 6 | 1 | 5 | 1 | 4 |

ADGLNPS | 7 | 2 | 0 | 3 | 2 | 6 | 1 | CDHJMPS | 2 | 2 | 1 | 3 | 3 | 7 | 4 |

ADGLOPS | 3 | 7 | 0 | 2 | 5 | 3 | 6 | CDHJNPS | 1 | 2 | 1 | 2 | 4 | 1 | 3 |

ADHJMPS | 4 | 3 | 3 | 3 | 1 | 2 | 2 | CDHJOPS | 1 | 3 | 3 | 2 | 2 | 3 | 3 |

ADHJNPS | 2 | 1 | 2 | 3 | 5 | 5 | 4 | CDHKMPS | 1 | 2 | 4 | 2 | 2 | 1 | 4 |

ADHJOPS | 4 | 2 | 4 | 4 | 3 | 5 | 1 | CDHKNPS | 2 | 7 | 3 | 2 | 8 | 2 | 1 |

ADHKMPS | 6 | 3 | 2 | 5 | 5 | 2 | 3 | CDHKOPS | 1 | 10 | 4 | 1 | 5 | 6 | 3 |

ADHKNPS | 5 | 4 | 1 | 1 | 4 | 2 | 3 | CDHLMPS | 3 | 4 | 3 | 6 | 3 | 1 | 1 |

ADHKOPS | 4 | 4 | 1 | 4 | 1 | 4 | 2 | CDHLNPS | 3 | 5 | 2 | 2 | 3 | 1 | 2 |

ADHLMPS | 7 | 3 | 3 | 1 | 1 | 4 | 2 | CDHLOPS | 4 | 3 | 5 | 3 | 4 | 4 | 2 |

ADHLNPS | 4 | 1 | 2 | 5 | 1 | 3 | 2 | CEGJMPS | 4 | 4 | 2 | 1 | 5 | 5 | 3 |

ADHLOPS | 2 | 5 | 0 | 1 | 1 | 7 | 4 | CEGJNPS | 2 | 1 | 4 | 2 | 3 | 3 | 8 |

AEGJMPS | 3 | 6 | 5 | 0 | 2 | 3 | 6 | CEGJOPS | 1 | 1 | 1 | 1 | 2 | 3 | 4 |

AEGJNPS | 2 | 4 | 3 | 1 | 2 | 4 | 4 | CEGKMPS | 2 | 4 | 3 | 7 | 4 | 2 | 2 |

AEGJOPS | 5 | 3 | 1 | 6 | 2 | 0 | 3 | CEGKNPS | 2 | 4 | 1 | 1 | 0 | 4 | 3 |

AEGKMPS | 5 | 1 | 3 | 6 | 2 | 4 | 1 | CEGKOPS | 1 | 2 | 1 | 3 | 1 | 4 | 4 |

AEGKNPS | 4 | 4 | 2 | 2 | 4 | 2 | 3 | CEGLMPS | 3 | 4 | 1 | 2 | 4 | 3 | 4 |

AEGKOPS | 7 | 4 | 3 | 1 | 0 | 2 | 2 | CEGLNPS | 5 | 3 | 2 | 5 | 4 | 4 | 5 |

AEGLMPS | 6 | 2 | 6 | 2 | 3 | 4 | 0 | CEGLOPS | 4 | 5 | 3 | 2 | 2 | 2 | 3 |

AEGLNPS | 3 | 5 | 1 | 3 | 2 | 2 | 3 | CEHJMPS | 1 | 3 | 1 | 3 | 3 | 2 | 1 |

AEGLOPS | 2 | 4 | 1 | 4 | 2 | 3 | 4 | CEHJNPS | 3 | 4 | 3 | 5 | 3 | 4 | 3 |

AEHJMPS | 2 | 3 | 5 | 0 | 2 | 3 | 1 | CEHJOPS | 4 | 4 | 3 | 3 | 2 | 6 | 3 |

AEHJNPS | 9 | 3 | 0 | 3 | 5 | 0 | 3 | CEHKMPS | 2 | 2 | 3 | 3 | 1 | 1 | 2 |

AEHJOPS | 5 | 4 | 4 | 3 | 0 | 5 | 3 | CEHKNPS | 2 | 5 | 3 | 2 | 4 | 3 | 0 |

AEHKMPS | 4 | 3 | 3 | 6 | 0 | 1 | 4 | CEHKOPS | 3 | 1 | 2 | 2 | 0 | 3 | 7 |

AEHKNPS | 4 | 1 | 5 | 6 | 3 | 3 | 6 | CEHLMPS | 2 | 4 | 4 | 1 | 6 | 5 | 3 |

AEHKOPS | 2 | 3 | 2 | 4 | 7 | 3 | 1 | CEHLNPS | 3 | 3 | 1 | 3 | 6 | 5 | 9 |

AEHLMPS | 1 | 1 | 2 | 2 | 4 | 5 | 0 | CEHLOPS | 5 | 3 | 3 | 3 | 3 | 4 | 4 |

AEHLNPS | 4 | 1 | 4 | 3 | 3 | 5 | 3 | NE | ADHKMPS; CDGLMPS; CEGLNPS; CDGKOPS | ||||||

AEHLOPS | 4 | 0 | 2 | 3 | 3 | 4 | 5 | Strat | {{A,B,C},{D,E},{G,H},{J,K,L},{M,N,O},{P},{S}} | ||||||

BDGJMPS | 3 | 2 | 5 | 2 | 4 | 3 | 4 | | |||||||

BDGJNPS | 2 | 4 | 2 | 3 | 2 | 4 | 1 | | |||||||

BDGJOPS | 4 | 2 | 1 | 3 | 0 | 1 | 2 | ||||||||

BDGKMPS | 2 | 8 | 3 | 1 | 2 | 0 | 4 | ||||||||

BDGKNPS | 2 | 4 | 3 | 3 | 2 | 0 | 2 | ||||||||

BDGKOPS | 2 | 2 | 3 | 3 | 3 | 3 | 0 | ||||||||

BDGLMPS | 4 | 2 | 3 | 4 | 4 | 4 | 6 | ||||||||

BDGLNPS | 5 | 8 | 6 | 2 | 6 | 5 | 6 | ||||||||

BDGLOPS | 2 | 2 | 8 | 4 | 3 | 5 | 5 | ||||||||

BDHJMPS | 1 | 1 | 2 | 3 | 4 | 3 | 5 | ||||||||

BDHJNPS | 3 | 1 | 1 | 4 | 1 | 4 | 5 | ||||||||

BDHJOPS | 2 | 5 | 3 | 3 | 3 | 3 | 2 | ||||||||

BDHKMPS | 3 | 5 | 4 | 4 | 5 | 1 | 4 | ||||||||

BDHKNPS | 5 | 4 | 2 | 0 | 3 | 4 | 2 | ||||||||

BDHKOPS | 2 | 4 | 2 | 3 | 2 | 4 | 4 | ||||||||

BDHLMPS | 3 | 2 | 1 | 2 | 3 | 4 | 3 | ||||||||

BDHLNPS | 6 | 3 | 5 | 2 | 2 | 2 | 1 | ||||||||

BDHLOPS | 2 | 4 | 2 | 8 | 3 | 2 | 5 | ||||||||

BEGJMPS | 2 | 3 | 4 | 7 | 1 | 3 | 2 | ||||||||

BEGJNPS | 5 | 2 | 4 | 3 | 4 | 3 | 3 | ||||||||

BEGJOPS | 4 | 4 | 2 | 5 | 3 | 1 | 3 | ||||||||

BEGKMPS | 4 | 2 | 1 | 1 | 1 | 6 | 4 | ||||||||

BEGKNPS | 2 | 5 | 3 | 1 | 3 | 2 | 2 | ||||||||

BEGKOPS | 4 | 2 | 4 | 5 | 5 | 1 | 1 | ||||||||

BEGLMPS | 6 | 2 | 5 | 1 | 3 | 2 | 3 | ||||||||

BEGLNPS | 2 | 5 | 2 | 0 | 1 | 1 | 6 | ||||||||

BEGLOPS | 4 | 1 | 1 | 7 | 3 | 4 | 6 | ||||||||

BEHJMPS | 1 | 3 | 1 | 2 | 0 | 4 | 3 | ||||||||

BEHJNPS | 3 | 3 | 3 | 2 | 1 | 7 | 3 | ||||||||

BEHJOPS | 4 | 1 | 3 | 2 | 4 | 2 | 4 | ||||||||

BEHKMPS | 2 | 2 | 3 | 1 | 2 | 3 | 5 | ||||||||

BEHKNPS | 3 | 6 | 4 | 1 | 4 | 4 | 1 | ||||||||

BEHKOPS | 1 | 8 | 3 | 4 | 5 | 5 | 2 | ||||||||

BEHLMPS | 6 | 6 | 2 | 5 | 1 | 3 | 4 | ||||||||

BEHLNPS | 3 | 6 | 3 | 5 | 1 | 4 | 2 | ||||||||

BEHLOPS | 3 | 1 | 1 | 2 | 3 | 3 | 3 | ||||||||

CDGJMPS | 2 | 3 | 4 | 1 | 3 | 5 | 3 | ||||||||

CDGJNPS | 2 | 3 | 2 | 3 | 3 | 3 | 4 |

The strategy set is {{A,B,C},{D,E},{G,H},{J,K,L},{M,N,O},{P},{S}}. The solutions for the 50 games are:

**Full dataset** is provided online. Note some sets are empty, which means at least a mixed-strategy equilibrium. Recall *games are restricted to pure-strategy play by assumption*. Such games are however common in the literature with countless applications. A Footnote on a mixed-strategy solver will accompany these findings in the future.

# CONCLUDING REMARKS 1

**g[]** deals with finite pure-strategy games. The solutions of 50 games were computed instantly with obsolete hardware. Thus, students or researchers with limited resources also benefit from this software. The statistical approach taken so far does not only safeguard code, but provides glimpses of future venues for research. Regularities such as the number of equilibria and their properties, given distributions underpinning incentives, is an arena for statistical inference and analysis. Huge games can be predicted or explored in this manner. Such feats are relevant in applications for purpose of institutional design, theory creation, and hypothesis testing. Statistical models and predictions about how many equilibria are to be expected given distributions underpinning incentives are treated in *Statistical Properties of NE*.

# EVOLUTIONARY GAME THEORY g[]

More realistic assumptions concerning the behavioural dispositions of isolated individuals and groups have gained traction over the years. One of the reasons is quite fierce critique of perfect rationality, especially unrealistic calculation prowess, presumably envisaged by Nash or Von Neuman. However, the duality between equilibria reached by perfectly rational individuals on one hand; and myopic ones through trial and error on the other, was emphasised from the outset. Such remarks remain relevant to understand one of the more powerful aspects of Game Theory. As I have argued elsewhere, multiplicity of equilibria can be used for purposes of institutional design and, among other things, be conceptualised as expressions of intent from a rational planer.

Moreover, critique of systems and outdated theory in their defence is not aided by discarding this link by means of hand waving. This connection provides clues to why systems prevail, in particular why flawed apologetic theories persist. In this setting myopic individuals converge to an important set of NE, guided by trial and error. Furthermore, more realistic assumptions about the psychology of individuals and non-pecuniary motives may have negligible impact on patterns of interaction and final outcomes, when not compatible with overarching economic structures. Nevertheless, it is easy to introduce them when employing universal solvers.

**Replicator dynamics **starts with a population consisting of n different types. The shares of the population playing (being) one of these types is denoted x_{i, }and expressed in the vector **x** containing the population distribution of these shares. The evolution of **x **is given by the following n-1 differential equations *at any point of time*

x’_{i }= x_{i}**[**π(s_{i},**x**) – π(**x**)**]**

These state that the share of the population using a particular strategy/being a type is determined by the difference between the payoff of such strategy π(s_{i},**x**) and the average payoff in the population π(**x**). The former payoff simply is the expected value of s_{i} given the population distribution **x**. The latter is computed from the former by taking the expected value of π(s_{i},**x**) instead. By convention, time (t) is suppressed in notation.

This two-player setting can be interpreted as follows: a pure strategy s_{i }of player one gives rise to a payoff (π) which reflects how it fares against nature. The latter plays some type with certain probability reflecting the population distribution. Therefore, symmetric matrices are readily consistent with this setup. Note however, that types can be interpreted as subsets of one entity.

**The evolutionary game** is constructed as follows. A symmetric nxn matrix (n=50) is generated with a **Poisson[3]** distribution* until a symmetric solution emerges*. Payoffs when two of the same type meet are the same for both. A game of this size is costly to generate with correct distribution and solutions, without a device such as **g[]**. This holds trueeven with a corresponding system of replicator-dynamics differential equations. The existence of symmetric NE must still be confirmed with such roundabout way, which in effect becomes taxing constraint in terms of time and computation.

Symmetric pure-strategy NE are fixed point in the corresponding Replicator-Dynamics systems for n>2. This setup and result is common-knowledge within Evolutionary Game Theory. Hence, the reader is encouraged to consult the mathematical research literature on this topic if needed.

# RESULTS 2

Data for this experiment is provided at online. There are three pure-strategy NE. One chooses 22 and the other 29; or both choose strategy 37.

**In the first experiment**, the latter symmetric NE is given a share of x_{37}(0)= 99%. The rest are given the uniform distribution[2]. As figure 1 shows, there is immediate convergence and stability throughout t ** [**0,500**]**. This system would have reached steady state fast if simulation ruled out negative values, i.e. in any conceivable (non-subjective) realistic setting interpreting x_{i} as shares in **[**0,1**]**. But the results are nevertheless strengthened as such detours may induce drifts, which can make the NE drop to zero, if the mean becomes negative. Otherwise, once a share reaches zero it stays there. In effect, the absence of a **[**0, 1**]** restriction makes the stability test more demanding.

**F1. Myopic Evolution & Symmetric NE (X _{37})**

The state vector **x **evolves to a homogenous population with a single type, namely x_{37 }which is the symmetric pure-strategy Nash Equilibrium of the corresponding 50×50 *static game*.

**The second experiment **searches for a lower bound for stability. At x_{37}(0) 95 %, and the rest uniformly distributed, the equilibrium holds until t=120. At x_{37}(0) 4/5 there is an initial convergence, followed by a sharp decline towards zero at t **[**24, 29**]**.

Instead it is x_{36} which stabilises between** [**1/5, 2/5**]**, suggesting a polymorphic equilibrium, which may correspond to a mixed-strategy NE. Analysis of such cases is postponed until the mixed-strategy solver is presented.

**F2. Stability Period & Initial Conditions**

**Clearly**, steady-state time increases dramatically about a 95 % threshold.

**The third experiment** takes a random sample of size 10 from the set of types and checks convergence with initial conditions x_{i}(0) 99%, and the rest uniformly distributed as above. The favoured types are

{18, 43, 35, 33, 23, 17, 42, 14, 46, 32}

None of the types in the sample, which consists of 1/5 of the total, are stable. Oddly enough, x_{37 }increases and approaches 1 when x_{17 }is favoured. Table 3 below shows these in groups.

Counting begins at the first column and proceeds downwards, i.e. 18 is in column 1, row 1; and 17 in column 2, row 1. No stable NE-convergence detected at t = 500.

**F3. 10 Favoured & Stability**

The reader is encouraged to consider a more analytical route, e.g. linearisation, or simulate the full sample of alternatives to the symmetric NE.

# CONCLUDING REMARKS 2

This section utilised a bridge between Evolutionary Game Theory and symmetric NE. This take also highlights a link between dynamics with myopic agents on one hand; and static one-shot games, with equilibria sustained by perfectly rational players, on the other. In doing so, I have provided verifiable information of a universal solver without disclosing code.

Replicator Dynamics confirmed the stability of a symmetric pure-strategy equilibrium, and the instability of others. Allowing negative state-variables of type distributions is unrealistic but makes the result more robust. In essence this relaxation of restrictions works as noise which perturb equilibria. Notwithstanding, there is a sharp exponential increase in steady-state time when the symmetric NE is given a share above 95%. A methodological point is made. Useful inferences about application of a class of models may be drawn from a particular conceptually flawed one, effectively unsuitable for realistic applications, at a fundamental mathematical level. Of course, it is not unthinkable idiosyncratic beliefs assign negative probabilities or shares.

[1] Coded by me at the outset this side-project, starting April 2022. I want students to learn and create with the principles of social science by constructing, and using, universal solvers.

[2] The uniform distribution is adapted to the initial condition of the NE type. Each type’s initial condition is random e.g. x_{i }~ U[0,1/a], and a must be s.t. **x **satisfies a certain Kolmogorov condition.