Hi everyone,

My latest game theory research follows below. In this case, it turned out to be rather simple, but interesting enough. It is about how a particular kind of strategy can be characterized with "winning streak budget constraints," essentially as a one-player game that ignores other players' strategies, given situations where players' choices do not depend on other players' "secret information about choices they've made."

An interesting example to consider: A generalization of chess where there is no checkmate, but instead only a final evaluation, after a certain number of moves, based on material gains against the opponent. In real-life chess, this might have an application to "the pace of obtaining an advantage against your opponent in chess."

It is also related to ideas about investment and real-world economics. Although winning-streak strategies are interesting in poker, poker is technically a coordination game (see below) so the theorem doesn't definitely apply--it isn't clear that it is a theta game. That is an example.

-Philip White (philipjwhite@yahoo.com)

Definition 1. Coordination game.

A coordination game G_0, which in this case we take to mean really either a coordination or anti-coordination game in the conventional sense (think of coordination games like "stag hunt" and anti-coordination games like "chicken") is any finite extensive form m-person complete-information game where there exist players P, Q, such that there exist decision nodes C_p, C_q, such that C_q is a descendant choice node of C_p in the extensive form game tree, C_p is a choice made by player P, and C_q is a node that represents a choice to be made by player Q such that this node is in an information set that implies that if the player Q is in that information set, Q does not know that player P has chosen C_p, and the impact of player Q's choice at C_q on the payoff of player Q is contingent on whether or not player P chose C_p or not; that is, if Q chooses C_q, it is possible that Q's final payoff will be different depending on whether P chose C_p or a different node at that point in the game. Note, we assume a pure-strategy paradigm here, but the principle is relevant to mixed-strategy-paradigm interpretations of the game, also. To put it another way: Essentially, a coordination game as we define it is any conventional game that is a game where at least one strategic actor has to make a decision based on a previous "unknown choice" of another strategic actor during the game.

Definition 2. Every-node-payoff extensive-form game.

In a conventional extensive-form game H, we can re-write this game so that the "optimal worst-case terminal payoff" at any given node is attained at any node where it can be first guaranteed that the strategic actor entering a particular node will obtain it as a minimum. Note, under this paradigm, we do not ever have any player decreasing his/her payoff at any node; the starting payoff at an initial node can be negative. Also note, we could allow "payoff losses" at certain nodes where lost payoff can be re-gained at other nodes, but this is not how we choose to express this definition in this case, although it is likely a useful alternate definition.

Definition 3. Winning streak budget constraints.

A winning streak budget constraint B is any expressed strategic goal of a player in a conventional (finite, m-person, extensive-form, complete-information) every-node-payoff extensive-form game where a player strives to achieve a certain payoff increase over the course of a certain number of "turns," where we consider how many moves the player has made after a particular initial node, and the net payoff accrued after the allocated number of "turns" has been taken by that player. We say that a budget constraint is "satisfied" if the player accrues at least the specified amount of utility points as payoff during that set of moves strategic moves by the player in the game.

Definition 4. C.

C = the union of the set of all coordination games and the set of all anti-coordination games.

Definition 5. Theta game.

A theta game is any finite m-person complete information game in which any player in the game may fully articulate his/her dominant pure strategy, such that the strategy cannot be improved upon under any condition by changing it to a different mixed or pure strategy, using a finite set of winning streak budget constraints, where the player's only task in the game is to make choices that will cause the budget constraints to be met or exceeded within each set of consecutive moves specified in the constraint. Note, this is an alternative to traditional expressions of strategies as contingency plans, and in this case, the strategy of each player can be written as a "full plan that does not take into account other players' strategic actions." That is, each player can strategize "on his/her own." Also note, in the case of conflicting specified winning streak budget constraints, in which a choice that helps to satisfy one constraint violates another, the strategic actor, after having accepted all of the constraints, may choose when playing the game to move in any way that does not violate *all* winning-streak budget constraints. (Strategic actors in practice should strive to select constraints of that nature that do not create final-payoff problems if such constraints, once followed as a "roadmap" during the game, have been adopted by a particular strategic actor.)

Definition 6. T.

T = the set of all finite m-person complete-information extensive form games that are theta games.

Definition 7. G.

G = the set of all finite m-person complete-information extensive form games.

Definition 8. W.

W = the set of all Walrasian games, as described on page 547 of Mas-Collel, Whinston, Green--here, we mean an ordered tuple consisting of: a private ownership economy, an allocation, and a price vector, where the allocation is what represents the "game-theoretic choices" of the strategic actors in the tuple, which can be re-expressed as a finite m-person extensive-form complete-information game with terminal payoffs indicated for all actors who chose their allocations.

Proposition 1. G \ C is a proper subset of T.

Proof. In every game that is not a coordination game as defined above, the game can be reduced to a one-player game for any fixed arbitrary player in the game--since no player's choice ever depends, final-payoff-wise in particular, on any other player's choice, by our definition of coordination games--in which any strategy that each individual player plays in their one-player game that is not dominated by any other strategy is an optimal strategy. Thus, each player seeking optimal payoff can merely set their "winning streak budget constraint" to be one that considers the collection of all moves, consecutively, throughout the game as the "streak," and the payoff of any non-dominated (pure, since it always can be) strategy as a constraint. Then, the strategic actor simply chooses choices that do not violate the constraint at any time.

In the other direction, we must prove that it is not the case that every theta game is a game that is not a coordination game. We seek to show that if each player in the game can specify his/her full strategy in a way that leads to optimal payoff, that the game is not necessarily not a coordination game, i.e., it can sometimes still be a coordination game. To do this, we take the contraposition of the contrary of our claim, that T is not a subset of G \ C, and assume it for the sake of contradiction. That is, we assume that if a fixed arbitrary game is a coordination game, then it is not a theta game where strategies can be expressed using winning-streak budget constraints. Since we have made no assumption about player rationality other than an interest in maximizing payoff--we do not have Nash-equilibrium-rationality assumptions relating to best responses here--we have that if a player A modifies his/her "secret choice" in advance of another player B's choice in any coordination game, then B's choice will affect B's payoff. However, we can devise a coordination game that is a theta game. The example we consider is "Prisoner's Dilemma with Room Selection." In this case, we have a traditional simultaneous-move Prisoner's Dilemma Game, expressed in extensive form with information sets, where the two prisoners, Prisoner A and Prisoner B, have one extra strategic choice to make in the game tree: Prisoner A chooses which room the warden (who is not a strategic actor in the game) will meet with Prisoner B in. The two rooms are labeled 1 and 2; if Prisoner B disobeys Prisoner A's choice, then Prisoner B receives the lowest possible payoff, lower than at all other terminal nodes. If Prisoner B agrees to meet with the warden to make a choice in whichever of the two rooms A designated, then B receives the traditional amount of payoff as in the conventional version of the Prisoner's Dilemma. Thus, in this case, Prisoner B's choice of where to meet the warden is contingent, terminal-payoff-wise, on a choice made by Prisoner A. At the same time, Prisoner B can still use a "winning-streak budget constraint" strategy as a guide to make the right choices at each decision node; the strategy would simply be to never make a move that jeopardizes the "Follow Prisoner A's Room Selection ; Defect" payoffs that Prisoner B would expect to be able to receive from playing that strategy. Due to the structure of the payoff matrix of this game, it is clear that no matter what Prisoner A's choice is, Prisoner B choosing to defect is the dominant strategy, so there is no question that striving for a certain minimal payoff will help here, and, given proper values in the payoff matrix, guide Prisoner B to choose to defect based on the winning-streak budget. Thus, we have a contradiction, and therefore as a result our full theorem.

On Wednesday, September 21, 2022 at 1:46:10 PM UTC-4, B.H. wrote:
> Hi everyone,
>
> My latest game theory research follows below. In this case, it turned out to be rather simple, but interesting enough. It is about how a particular kind of strategy can be characterized with "winning streak budget constraints," essentially as a one-player game that ignores other players' strategies, given situations where players' choices do not depend on other players' "secret information about choices they've made."
>
> An interesting example to consider: A generalization of chess where there is no checkmate, but instead only a final evaluation, after a certain number of moves, based on material gains against the opponent. In real-life chess, this might have an application to "the pace of obtaining an advantage against your opponent in chess."
>
> It is also related to ideas about investment and real-world economics. Although winning-streak strategies are interesting in poker, poker is technically a coordination game (see below) so the theorem doesn't definitely apply--it isn't clear that it is a theta game. That is an example.
>
> -Philip White (philip...@yahoo.com)
>
>
>
>
>
>
>
> Definition 1. Coordination game.
>
> A coordination game G_0, which in this case we take to mean really either a coordination or anti-coordination game in the conventional sense (think of coordination games like "stag hunt" and anti-coordination games like "chicken") is any finite extensive form m-person complete-information game where there exist players P, Q, such that there exist decision nodes C_p, C_q, such that C_q is a descendant choice node of C_p in the extensive form game tree, C_p is a choice made by player P, and C_q is a node that represents a choice to be made by player Q such that this node is in an information set that implies that if the player Q is in that information set, Q does not know that player P has chosen C_p, and the impact of player Q's choice at C_q on the payoff of player Q is contingent on whether or not player P chose C_p or not; that is, if Q chooses C_q, it is possible that Q's final payoff will be different depending on whether P chose C_p or a different node at that point in the game. Note, we assume a pure-strategy paradigm here, but the principle is relevant to mixed-strategy-paradigm interpretations of the game, also. To put it another way: Essentially, a coordination game as we define it is any conventional game that is a game where at least one strategic actor has to make a decision based on a previous "unknown choice" of another strategic actor during the game.
>
>
> Definition 2. Every-node-payoff extensive-form game.
>
> In a conventional extensive-form game H, we can re-write this game so that the "optimal worst-case terminal payoff" at any given node is attained at any node where it can be first guaranteed that the strategic actor entering a particular node will obtain it as a minimum. Note, under this paradigm, we do not ever have any player decreasing his/her payoff at any node; the starting payoff at an initial node can be negative. Also note, we could allow "payoff losses" at certain nodes where lost payoff can be re-gained at other nodes, but this is not how we choose to express this definition in this case, although it is likely a useful alternate definition.
>
>
> Definition 3. Winning streak budget constraints.
>
> A winning streak budget constraint B is any expressed strategic goal of a player in a conventional (finite, m-person, extensive-form, complete-information) every-node-payoff extensive-form game where a player strives to achieve a certain payoff increase over the course of a certain number of "turns," where we consider how many moves the player has made after a particular initial node, and the net payoff accrued after the allocated number of "turns" has been taken by that player. We say that a budget constraint is "satisfied" if the player accrues at least the specified amount of utility points as payoff during that set of moves strategic moves by the player in the game.
>
>
> Definition 4. C.
>
> C = the union of the set of all coordination games and the set of all anti-coordination games.
>
>
> Definition 5. Theta game.
>
> A theta game is any finite m-person complete information game in which any player in the game may fully articulate his/her dominant pure strategy, such that the strategy cannot be improved upon under any condition by changing it to a different mixed or pure strategy, using a finite set of winning streak budget constraints, where the player's only task in the game is to make choices that will cause the budget constraints to be met or exceeded within each set of consecutive moves specified in the constraint. Note, this is an alternative to traditional expressions of strategies as contingency plans, and in this case, the strategy of each player can be written as a "full plan that does not take into account other players' strategic actions." That is, each player can strategize "on his/her own." Also note, in the case of conflicting specified winning streak budget constraints, in which a choice that helps to satisfy one constraint violates another, the strategic actor, after having accepted all of the constraints, may choose when playing the game to move in any way that does not violate *all* winning-streak budget constraints. (Strategic actors in practice should strive to select constraints of that nature that do not create final-payoff problems if such constraints, once followed as a "roadmap" during the game, have been adopted by a particular strategic actor.)
>
>
> Definition 6. T.
>
> T = the set of all finite m-person complete-information extensive form games that are theta games.
>
>
> Definition 7. G.
>
> G = the set of all finite m-person complete-information extensive form games.
>
>
> Definition 8. W.
>
> W = the set of all Walrasian games, as described on page 547 of Mas-Collel, Whinston, Green--here, we mean an ordered tuple consisting of: a private ownership economy, an allocation, and a price vector, where the allocation is what represents the "game-theoretic choices" of the strategic actors in the tuple, which can be re-expressed as a finite m-person extensive-form complete-information game with terminal payoffs indicated for all actors who chose their allocations.
>
>
> Proposition 1. G \ C is a proper subset of T.
>
> Proof. In every game that is not a coordination game as defined above, the game can be reduced to a one-player game for any fixed arbitrary player in the game--since no player's choice ever depends, final-payoff-wise in particular, on any other player's choice, by our definition of coordination games--in which any strategy that each individual player plays in their one-player game that is not dominated by any other strategy is an optimal strategy. Thus, each player seeking optimal payoff can merely set their "winning streak budget constraint" to be one that considers the collection of all moves, consecutively, throughout the game as the "streak," and the payoff of any non-dominated (pure, since it always can be) strategy as a constraint. Then, the strategic actor simply chooses choices that do not violate the constraint at any time.
>
> In the other direction, we must prove that it is not the case that every theta game is a game that is not a coordination game. We seek to show that if each player in the game can specify his/her full strategy in a way that leads to optimal payoff, that the game is not necessarily not a coordination game, i.e., it can sometimes still be a coordination game. To do this, we take the contraposition of the contrary of our claim, that T is not a subset of G \ C, and assume it for the sake of contradiction. That is, we assume that if a fixed arbitrary game is a coordination game, then it is not a theta game where strategies can be expressed using winning-streak budget constraints. Since we have made no assumption about player rationality other than an interest in maximizing payoff--we do not have Nash-equilibrium-rationality assumptions relating to best responses here--we have that if a player A modifies his/her "secret choice" in advance of another player B's choice in any coordination game, then B's choice will affect B's payoff. However, we can devise a coordination game that is a theta game. The example we consider is "Prisoner's Dilemma with Room Selection." In this case, we have a traditional simultaneous-move Prisoner's Dilemma Game, expressed in extensive form with information sets, where the two prisoners, Prisoner A and Prisoner B, have one extra strategic choice to make in the game tree: Prisoner A chooses which room the warden (who is not a strategic actor in the game) will meet with Prisoner B in. The two rooms are labeled 1 and 2; if Prisoner B disobeys Prisoner A's choice, then Prisoner B receives the lowest possible payoff, lower than at all other terminal nodes. If Prisoner B agrees to meet with the warden to make a choice in whichever of the two rooms A designated, then B receives the traditional amount of payoff as in the conventional version of the Prisoner's Dilemma. Thus, in this case, Prisoner B's choice of where to meet the warden is contingent, terminal-payoff-wise, on a choice made by Prisoner A. At the same time, Prisoner B can still use a "winning-streak budget constraint" strategy as a guide to make the right choices at each decision node; the strategy would simply be to never make a move that jeopardizes the "Follow Prisoner A's Room Selection ; Defect" payoffs that Prisoner B would expect to be able to receive from playing that strategy. Due to the structure of the payoff matrix of this game, it is clear that no matter what Prisoner A's choice is, Prisoner B choosing to defect is the dominant strategy, so there is no question that striving for a certain minimal payoff will help here, and, given proper values in the payoff matrix, guide Prisoner B to choose to defect based on the winning-streak budget.. Thus, we have a contradiction, and therefore as a result our full theorem..
>
>
> Conjecture 1. W is a subset of T.
>
> No proof supplied at this time.

On 9/21/2022 10:46 AM, B.H. wrote:
>
> Hi everyone,
>
> My latest game theory research follows below. In this case, it turned out to be rather simple, but interesting enough. It is about how a particular kind of strategy can be characterized with "winning streak budget constraints," essentially as a one-player game that ignores other players' strategies, given situations where players' choices do not depend on other players' "secret information about choices they've made."

Hello!

if it is a one-player game, there are no other players!

Goodbye!

On Wednesday, September 21, 2022 at 9:09:36 PM UTC-4, Eduardo Fahqtardo wrote:
> On 9/21/2022 10:46 AM, B.H. wrote:
> >
> > Hi everyone,
> >
> > My latest game theory research follows below. In this case, it turned out to be rather simple, but interesting enough. It is about how a particular kind of strategy can be characterized with "winning streak budget constraints," essentially as a one-player game that ignores other players' strategies, given situations where players' choices do not depend on other players' "secret information about choices they've made."
> Hello!
>
> if it is a one-player game, there are no other players!
>
> Goodbye!

Yes, it is reducible to a one-player game, not exactly equivalent to one. The point is that the choices made by the individual player can be modeled as a one-player game, where choices made at any point do not depend on choices by other players from the past, and, as it turns out, where such choices also are not hidden from other players in the future, by definition. Thus, although prisoner's-dilemma-type-simultaneity can occur in the game--"simultaneous move" without consequence to the players' strategies--it is essentially equivalent to a perfect information game. Backwards induction can be used in that case to predict each player's choices completely. In the event of a player who has two choices that are equivalent, final-payoff-wise, to each other, and which choice is made impacts other players, the choice could be modeled as selected at random and controlled by Nature.

That is strange...my proof wasn't written out as thoroughly as I had thought. Admittedly, unlike crooked regimes in Israel and the US, I am not a torture psychologist. If highly invasive efforts are made to psychologically torture me, I cannot be held responsible for mistakes or sloppiness in work I am trying to do; torture is too much to expect any reasonable person to bear, even from the so-called inheritors of the court of the Holocaust--it's a title held by Israeli people but not its broken right-wing extremist government.

In this case, my proof was correct. I wrote it well enough; it's my para-journal in a sense anyway. If there had been a true mistake, the blame would lie with the people who tortured me; I have no ability to predict and understand what torture--being choked while sleeping in this case, to the point where my windpipe felt closed and I couldn't breathe and woke up and walked as quickly as possible to the bathroom to drink water which helped--does to my brain and my ability to produce top-quality work.

Fortunately I was still right this time; my intuition made up for the deficit in the precision of my explanation.

-Philip White (philipjwhite@yahoo.com)