Nash Equilibria <-> Coherent Cooperative Games
Stability amongst chaos
Pictures paint a thousand words. So when we picture Coherent Cooperative Games and Nash Equilibria, we picture stability within a chaotic system:
I define a Coherent Cooperative Game as one where:
“If you play, there is only one type of play or move which is that on each move you play the other player wins. The rules of the game are open and coherently known to each player. Play tokens are defined by the rules and may be the rules themselves.”
…and I have also said that a Coherent Cooperative Game may sound like the most boring game on Earth, where on each play, the other player/s win; meaning all players win.
On the face of it, it may sound as if completely uneventful, however when all players driving a car drive on the correct side of the road it does not mean that the music can’t be playing, people singing along, bobbing their head to the beat and playing I Spy as they drive along. Coherence and cooperation have their beauty in stability, equitability and surance in each player knowing and understanding the rules under which a Coherent Cooperative Game is played.
But not all systems are stable, as in road rules.
Prices fluctuate, wars break out, competition reigns, rules change, conceptions fluctuate, and accidents/misconceptions happen.
The whole idea of Coherent Cooperative Games (CCG) is that all players win, so finding that coherence is important, especially if driving on the road where we want, need and desire that all players drive on the correct side of the road.
I call a divergence from coherence under a CCG a Differential Interpretation Game. Within law, torts of law, negligent behaviour, criminal intent, misdemeanour, within international trade…unfair practices, price-gouging, product dumping, monopoly, oligopoly, within international relations…differences in opinions, culture, philosophy, religion etc.
Harmony, for instance, a CCG as when early Muslims and still the faithful supported and lived/worked along-side Christians. Or a balance of trade in international economics.
A Coherent Cooperative Game, as I describe them, neither altruism or stand-over/debt-bondage, but where all players happy with the equilibrium reached.
A Nash Equilibrium, a stable state within a simplex of choices, a happy medium where no rational player stands to gain by changing their choices (of play), is to my mind a ‘state of play’ within a grander Coherent Cooperative Game because life goes on and we don’t drive but once on the correct side of the road, we don’t buy milk once from the grocery store…and prices fluctuate (offering a differential interpretation) within which a new Coherent Cooperative Game must be reached if for there to be a Nash Equilibrium on each further play or role of the dice.
I.e. A Nash Equilibrium more-or-less defines a point in time, whereas a Coherent Cooperative Game puts things in context of reaching Nash Equilibria among players, within a potentially and otherwise chaotic system, which on the one side would otherwise be anarchy to the n-th degree (no rules and no stable state/s), warfare (non-cooperative games) and on the other side stasis or stagnation with no interoperability or further play.
So…Nash Equilibria are important, but must be seen in the context of a Coherent Cooperative Game, because even though Nash Equilibria born from non-cooperative games, if an equilibrium exists at all, then each player, at least in one point in time, must have agreed on something if the Nash Equilibrium executed upon.
More pointedly…Nash proved that a Nash Equilibria exist within simplexes of choices, not that any player actually ever agree on taking up the best option.
So, the way I see it…if players choose to win, and can accept that other players are choosing to win, and by each player playing such that the other player win, then there exists a Coherent Coopertaive Game on the offering.
Don’t be fooled however. Players need not choose to play a CCG, as it stands with a very big IF:
“IF you play, there is only one type of play or move which is that on each move you play the other player wins. The rules of the game are open and coherently known to each player. Play tokens are defined by the rules and may be the rules themselves.”
In this way…Coherent Cooperative Games offer a way to conflict resolution.
Solution/Problem Space within the Simplex
When I first started studying Nash Equilibria in earnest about 14 years ago, I made my own simplex out of sticks and the problem/solution space a stretched stocking between points.
Enjoy as you find a Nash Equilibrium within a Coherent Cooperative Game and over a myriad of ways of looking at the same problem:
(Images by author)
Thanks for reading. As time permits, I will write more on Nash Equilibria and Coherent Cooperative Games.
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