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One of the common conceptions in AI is the idea of game theory. We see that in the predominance of chess and other games in the literature as metrics of AI success. We see it in the names of machine learning concepts such as generative adversarial networks. We see it in academia in that a limited number of seats in teaching and administrative structure are available. We see it in perceptions of biological history. We see it in sports and finance and geopolitical endeavors.

Is Our Current Perception Skewed?

When mathematician John von Neumann and economist Oskar Morgenstern wrote Theory of Games and Economic Behavior, it was 1943. Nazi forces were devastating Europe and Japanese forces had taken Korea and much of China. The context of games was intended to be economic, but von Neumann was also involved in the technological revolution needed to oppose these forces in an adversarial game called WWII. The idea of games changed from play to preservation of civilization.

Yet civilization, although it may rely on defensive action to counter what may or may not be destructive offense, it is more defined by collaboration for mutual benefit (Finnish win-win-tilanne). Civilization is a game of preservation and the avoidance of loss, as John Donne intoned in his famous poem when he wrote, "If a clod be washed away by the sea, Europe is the less," implying that the loss of one person is a loss for all, based on the Golden Rule (Jesus). It is this higher thinking of the objective of the game that preserves civilization, not adversarialism.

  • Lives
  • Families
  • Ideas

It is often the unique, original ideas that forward civilization in the area of ideas. It is often the lives and families of those who, at the time the idea arises, are not in power that change the thought collective (Ludwik Fleck). These are a few examples of those that disturbed cultural norms (including scientific ones) and led to a forward step for us all.

  • Socrates
  • Saul of Tarsus
  • Hypatia of Alexandria
  • Galileo
  • Isaac Newton
  • Huang Yuanyong (黃遠庸)
  • Mahatma Gandhi
  • Martin Luther King
  • Richard Stallman

Reconciliation of Game Theory with Concepts of Civilization

Nobel laureate John Forbes Nash Jr. contributed to the mathematical idea that in economic systems real world curvature in relations, the game is not a zero sum game. This is the basis for the question.

As gaming continues to penetrate further into mobile device and Internet usage and as AI games continue to gain degrees of angle in the pie chart of most national defense spending budgets, the definition of games that are not based on the WWII model of what a game must be gains importance.

How can designers of educational, entertainment centered, economic, financial, and military games transition their thinking toward cherishing alternative thought and re-prioritizing concepts like detente, playing for fun, employee collaboration, and educational objectives that are not zero sum GPA (grade point average) games based on #2 pencil multiple guess assessments?

Mathematics Leading to Game Design Concepts

First must come a mathematical idea of value, where the value between the individual and the body of all humans is tied. The mathematics may be theoretical (and probably should be at first), but must also be applied such that no one can achieve a high score in a game without developing an increase in the velocity of achievement in class of those with low achievement velocities. This requires a curved (non-linear) system, much like occurs in nature. It is a more realistic game than where all the adversaries are killed and the champion is now alone in the game space such that, had the game application not ended, no real value was gained — only loss.

Incentivization

In this context, we can return to the title question.

How can a collaboration game be defined mathematically?

What kinds of value functions incentivize collaboration and frustrate the actions of those that try to trounce everyone else to get ahead? How can we characterize the curvature of the required functions such that too many individual losses block the ultimate achievements of current leaders? If the Golden Rule, mutual appreciation, and collaboration are truly the ways civilization progresses forward, how can scoring be designed to reflect this reality? What qualities, mathematically, must the model have?


The objective is to define mathematically a very specific thing indicated in the title question. That some background is give is solely to encourage out of the box thinking in responding specifically and correctly to a narrow and well defined question.

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    $\begingroup$ The problem is not your questions--you ask great and salient questions. But you tend to do what I used to do which was present a thesis as a route to the ask. (I've limited that b/c my sense is people feel it's abuse of the Q&A structure.) My advice is present only the abstract in the question, no more than a few (ideally) short paragraphs, then link to a blog with the thesis (which puts some traffic on the blog.) I've also learned that when the question is really long, nobody reads it because we all 95% parse due to the information explosion. $\endgroup$ – DukeZhou Mar 14 at 23:20
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    $\begingroup$ Agreed with DukeZhou, in brief - xkcd.com/1640 - this question gives too much context and red-herring leads to the answers. It is not at all clear what a good answer would look like, and my immediate reaction as someone who could attempt to answer this is that you would not be satisfied unless I attempted to address everything with a question mark after it in the question body. Which is far too much work and would result in a rambling essay-style answer. Whilst a reference to difference between Pareto Optimality and Nash Equilibrium and methods to reconcile them could be useful. $\endgroup$ – Neil Slater Mar 15 at 9:59
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The initial problem may arise from overemphasis on "winner take all", specifically:

Under the normal play convention, output is boolean. T/F = Win/Lose. My sense is that there has been overfocus on binary output in combinatorial game theory and algorithmic CGT, possibly because normal play is attractive from a computability standpoint (greatly simplifies the mathematics.)

Elwyn Berelkamp wrote an interesting paper called The Economist’s View of Combinatorial Games which seems to comment on this.

Essentially, in scoring games, output is a ratio. You see this with Bridge and Go. Go demonstrates the overfocus on boolean output in that, in nearly all analysis, the outcome is reduced to Win/Lose.

In a foundational ACGT paper Constraint Logic: A Uniform Framework for Modeling Computation as Games, the authors assert that:

For any game, there is a standard decision question: does [a given player] have a forced win from a given position?

This is not incorrect, but demonstrates a limited frame of reference--viewing outcomes as binary and in reference only to the self (the "given player"). Another passage more clearly explicates the concept:

We consider the generalization to multiplayer games. It turns out that naıvely adding players beyond two does not increase the complexity of the standard decision question, “does player X have a forced win?”. We might as well assume that all the other players team up to beat X, in which case we effectively have a two-player game again.

  • The implication is that all other players are considered a coalition against player X.

However, such a coalition is meaningless if only a single player can "win".

Berelkamp nicely encapsulates what might be termed a Non-Classical Decision Problem:

At every position of a game such as Go or Domineering, there are two very important questions: Who is ahead, and by how much?

Berelkamp explicates further:

Following Conway [1976], classical abstract combinatorial game theorists answer these questions with a value and an incentive, as discussed inWinningWays [Berlekamp et al. 1982]. These answers are precisely correct when the objective of the game is to get the last legal move. Values and incentives are themselves games, and can quickly become complicated. Our ideal economist takes a different view. Following Hanner [1959] and Milnor [1953], he views the game as a contest to accumulate points, which can eventually be converted into cash.

Bear in mind we're still talking about 2-player games, but this model naturally extends to games with players > 2, and the results are quantitative, not boolean.

It is assumed that in multi-agent games, coalitions will naturally form among disadvantaged players to mitigate advantage of agents in stronger positions.

  • Coalitions naturally arise in games with players > 2.

Collaborative strategy arises naturally but isn't the answer you're looking for. That answer I believe can be found in traditional (economic) game theory, via Hofstadter and the concept of superrationality:

In economics and game theory, a participant is considered to have superrationality (or renormalized rationality) if they have perfect rationality (and thus maximize their own utility) but assume that all other players are superrational too and that a superrational individual will always come up with the same strategy as any other superrational thinker when facing the same problem. Applying this definition, a superrational player playing against a superrational opponent in a prisoner's dilemma will cooperate while a rationally self-interested player would defect. SOURCE: Superrationality (wiki)

In keeping with your own line of reasoning:

  • Renormalized rationality can be seen as a mathematical validation of the Golden Rule, which mitigates the "Iron Rule" of minimax and always defecting.

The caveat is that it does require willingness to sacrifice ("taking a hit" in the economic sense) and even "turning the other cheek" (willingness to take a second "hit" to give the competitor a chance to change their behavior.)

Presumably, this willingness to sacrifice explains why the notion of superrationality is so unpopular.

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My own work involves partisan sudoku as a non-classical decision problem, and provides a much more compact model for non-chance, perfect information scoring games, which can be extended into players > 2 with no mechanical alteration. The linked variant might be termed a "resource scheduling game", in keeping with the historical utility of Latin squares in scheduling problems. However, the model can either reduced to a simpler form of partisan sudoku, or extended to include imperfect information and "displacement" (conditions for removing tokens which would render the games cyclic and potentially non-finite.)

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  • $\begingroup$ The focus on binary output in combinatorial game theory makes sense, because everything can be described as zeros and ones. Zero means false and one is equal to true. This is the fundamental concept of computation and turing-machines. $\endgroup$ – Manuel Rodriguez Mar 14 at 21:04
  • $\begingroup$ @ManuelRodriguez yup, but in non-classical decision problems (scoring games) this reduction to boolean requires discarding information (the ratio that defines the score between p players.) It's possible here that adherence to a consistent but incomplete formal system is a root of the problem. $\endgroup$ – DukeZhou Mar 14 at 21:22
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There are three different kind of games available: single player games for example a puzzle in which one person has to take decisions. Multi-player games for example a real-time strategy game in which player1 is acting against player2 and collaboration games for example Wikipedia in which the users are teaching each other. The difference between single player games and multi-player games is not that hard. It's only a game with more users at the same time, but the game is played with fixed rules. The interesting aspect is the difference between multi-player games and collaboration games.

A collaboration game hasn't a fixed ruleset and the assumption is, that the actors are humans. A non-human entity in an online discussion forum will be detected as wrong. That means, a robot can't be part of a collaboration game. In general, it's about social games which are played in politics, schools, society, capitalism and online-forums. They have in contrast to Monopoly not a fixed ruleset but a fixed group behavior. Defining such kind of games mathematically is an open problem. Part of the equation is a real human or a machine acting with human-level-capabilities. The only thing what is possible with current technologies is the so called turing-test to determine if an actor in a collaboration game isn't human like.

Quote: “Turing’s criterion for cognition was individual, autonomous input/output capacity. It is not clear that distributed cognition could pass the Turing Test.” Harnad, Stevan, and Itiel E. Dror. "Distributed cognition: Cognizing, autonomy and the Turing Test." Pragmatics & Cognition 14.2 (2006): 209-213.

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  • $\begingroup$ To explain my downvote here, you make an unsupported assertion about games not reflected in the established literature. Specifically, ACGT defines zero player (cellular automata), one player (puzzles), and two players games. In partisan team games, each team would be regarded as a single player. In collaborative games were all players work together, they would be regarded as a single team, and the endeavor would be a one player game. $\endgroup$ – DukeZhou Mar 14 at 22:53
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    $\begingroup$ According to Algorithmic Combinatorial Game Theory, one player games are NP-complete. This definition makes only sense, if the aim is to use mathematics, to proof that Artificial Intelligence isn't practical. It think, it make sense, to strip down mathematics from game theory and treat games as simulation of reality. $\endgroup$ – Manuel Rodriguez Mar 15 at 7:39
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Weirdly I think you would benefit from analysing group power dynamics in gtao and rdr2o.

(Grand Theft Auto Online and Red Dead Redemption Online)

Both games require team work to be successful. Especially true of Gtao. Which has a power dynamic that allows one player with enough firepower to sabotage team efforts so analysing behavior of players and likelihood of success is important. It's obscure academically I know but given that a player can put in 10 hours work gain product for a business and lose it durimg the sell phase to other players fairly easily you need diplomacy, behavior analysis and a strong team to ensure success during collaborative play. Playing alone is unlikely to gain the win needed or keep a stable free roam environment that allows business activity.

In short, no online game currently has higher risk/ reward in collaborative play and it might give you some ideas on how to define your formula.

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