Hi everyone,

What follows is a great game theory idea related to political science that I thought of today. Publishing to Usenet is good enough for me; anyone mathematically literate enough will follow the ideas, and my audience is mainly international governments, businesses, and non-profits anyway. Game theory, unlike CS, is better when published; rational understanders of the ideas presented tend to adhere to good principles, which relate to "good strategy for everyone" as opposed to "secret processes for one party's competitive unique competitive advantage."

So here is my idea. The problem in question is the "dominant beneficiary backer problem"--when a powerful leader can hire representatives/"problem janitors" or jettison backer interests to strategically do upsetting things while evading personal or "close-by" accountability, other strategic actors may start to find this type of leader very troublesome and hard to deal with. Leaders who insist on "passing the buck" when challenged to other supporters, voluntarily or not, are difficult to pin down and stop from damaging one's interests.

Define an "accountability and support graph" as follows: a finite digraph with k nodes and edges between them has three node weights and one edge weight. The edge weights can be positive, negative, or zero; positive means "helpful support," negative means "damaging opposition," and zero means no interaction. The three node weights are: unaccountability support (essentially, a defense against damage from "damaging opposition" that causes the strategic actor to pass along "bad damage" by distributing the exact amount of damage to other actors that support it with a positive value; note, some of these strategic actors may have even higher "unaccountability support values," and may be able to pass their own problems along to a "broad-scale economy" actor that simply suffers and accepts the damage), power to support others, power to damage others.

At every time step of the graph, the strategic actors may increase or decrease their three positive node weights to any value within the "maximum" specified for each such value for that actor, and 0. Additionally, they may alter their outgoing edge weights to be any rational number with (let's say) 20 digits in the interval [-1,1]. Further, each node accrues and loses points in accordance with their support and damage received from other actors.. The raw total is the (support power)*(positive edge weight) from "helpful actors," + (damage power)*(negative edge weight), which is a non-positive quantity, from "damaging actors." Based on the raw score, each node/actor may also pass along up to the "unaccountability support value" of incoming negative damage (held as a separate quantity from support) and redistribute it, along the edges, to any actor who supports this actor with a positive value, regradless of the "magnitude" of that positive value. This redistribution continues until the entire graph has "settled" after however many repetitions, and all points for this turn have been calculated.

This game may be repeated for any positive finite natural number of cycles. The challenge, stated mathematically and in a general way, is: Given a collection of opponent strategies at a particular time step, or in the overall game with a finite number of time steps, how, given all of the strategies of the players, should a player proceed? Note that the strategies of the opponents might be irrational.

Here is the simple part: Simply find the best response. A while ago, I devised an excellent PTIME algorithm for finding Nash equilibria of finite m-person complete-information games, and this algorithm can be used to calculate best responses when players may not be playing rationally, too. The algorithm is simple: Just take each player's mixed strategy as a real-valued linear combination of ordered pairs of payoffs in a k-dimensional matrix. Then, a simple application of multivariable calculus techniques can be used to find all critical points, and thus all maxima and minima needed to establish optimal payoffs. Nash proved that a Nash equilibrium must exist; thus, all the mathematician must do is find any mixed strategy profile that represents a local maximum for all players. Any such strategy profile is a Nash equilibrium. Similarly, given other players' fixed strategies, a single remaining player's best-response mixed strategy can easily be identified using calculus; find the global maximum for that player's payoff, given the other players' fixed constant strategies.

The mere application of that algorithm to a reduction of the above graph-game to an extensive form and then payoff-matrix-form game is sufficient to help strategic actors that are able to model strategic interactions with such a digraph-based approach, preferably in a way where other strategic actors would agree with the exact stated graph, but even without such a condition--as long as the model is accurate and other players' strategies are specified completely, to find optimal strategic responses to difficult strategic situations that are related to accountability avoidance and re-distribution ("buck passing") and general ability to support and damage allies and adversaries.

I hope that my publication of this idea to Usenet will help the world become a more accountable and just place!

-Philip White (philipjwhite@yahoo.com)