Multi-Game Theory: Zero-Knowledge Strategies and Boundary Analysis
Game theory is an academic field of study. Here, “game” does not refer to video games or recreational activities but encompasses any structured activity in which participants make decisions based on rules.
Game theory typically analyzes the choices rational participants make given the specific rules of a game.
In reality, we engage in various types of games, many of which are repetitive.
Furthermore, it is impossible to know the entirety of the games we are involved in beforehand, as there is significant uncertainty. Such situations can be described as open-ended games.
When considering multiple types of games, repeated games, and open-ended games, rational decisions differ fundamentally from those made within the confines of a single game.
Therefore, instead of focusing solely on rational choices within individual games, it is crucial to analyze the entirety of multiple games.
This approach is referred to as multi-game theory.
Zero-Knowledge Multi-Game Strategies
In the context of multi-game theory, it is practically impossible to understand the rules of all games and analyze the optimal choices across the entire system.
As the number of games increases, complexity grows exponentially, making analysis within realistic time constraints unfeasible. Furthermore, since games are open-ended, the existence of unknown games inherently prevents exhaustive analysis.
Thus, an approach is needed that treats individual games abstractly, without detailed analysis of their rules, to analyze the entirety of multi-game systems.
When games are maximally abstracted, we reach a state where nothing is known about the rules or participants of each game. This is referred to as a zero-knowledge multi-game.
It might seem that no analysis can be performed when there is no knowledge of game rules or participants. However, intriguingly, zero-knowledge multi-games adhere to universal principles that enable universal analysis.
One such universal principle is that if all participants collaborate, the total payoff for all participants can often be maximized.
Of course, in single games where the total payoff does not vary regardless of the choice, collaboration does not affect the total payoff. However, in games where the total payoff varies based on choices, collaboration enables participants to aim for choices that maximize the total payoff.
In particular, when game rules are explicitly stated and there is no uncertainty, it is possible to reliably choose options that yield the highest total payoff.
Another universal principle of zero-knowledge multi-games is that when multiple games are considered together, the overall optimal choice is more likely to be identified compared to analyzing individual games in isolation.
Naturally, if the outcomes of choices in each game have no impact on others, combining games does not affect the results. However, when games influence one another, considering them together increases the likelihood of achieving optimal outcomes overall.
Basic Strategies for Multi-Games
Based on these observations, zero-knowledge multi-games reveal two universal principles:
First, obtaining collaboration from all participants is crucial.
Second, considering multiple games in combination rather than in isolation enhances the potential to optimize total payoffs.
These principles lead to a clear rational strategy for multi-games:
“Collaborate with everyone unless there is a reason not to, and make decisions by considering multiple games together.”
This universal strategy underscores the importance of focusing on methods to achieve collaboration and integrate multiple games before delving into the specifics of individual games.
In essence, the true challenge lies in identifying and eliminating the barriers that prevent collaboration or integrated consideration of games.
This is not an ethical argument advocating collaboration. Instead, in multi-games, constructing a collaborative framework is the most reliable and rational strategy for pursuing individual interests.
Barriers to Collaboration
Barriers that prevent collaboration or the integrated consideration of multiple games stem from concrete causes rather than abstract issues like inequality.
Specific factors such as the deprivation of participants’ minimum necessary resources or rights hinder collaboration.
High levels of uncertainty also act as a barrier. As uncertainty increases, refusing collaboration to secure certain gains becomes more rational.
Additionally, participants’ desires, such as the need for recognition or the wish to acquire more than others, can obstruct collaboration.
By focusing on these barriers and removing them, it becomes possible to achieve the optimal overall payoff in multi-games.
Strategies for Removing Barriers
The specific methods for removing barriers depend on the particular games and participants involved. Additionally, participants’ circumstances and the overall situation may change as games progress.
For this reason, it is not feasible to predetermine rigid methods. Dynamic adaptation is essential.
To achieve this, it is important to continuously identify, address, and improve the removal of barriers. Furthermore, this effort should not rely solely on a single individual but involve all participants collectively.
An ideal approach is to establish a mechanism that incentivizes participants who contribute to removing barriers.
This system leverages desires such as the need for recognition or the wish to gain more than others, redirecting those motivations toward removing barriers and achieving a dual benefit.
In essence, this approach prevents individual desires from hindering cooperation and the integration of multiple games, while simultaneously harnessing those desires to continually improve methods for removing barriers and adapting to changing circumstances.
Over time, this strengthens cooperation and the integration of multiple games, making them more robust.
For participants who understand multi-game theory, constructing a cooperative framework is rational, regardless of ethical considerations. Such participants are likely to proactively address and implement methods for removing barriers to cooperation.
For participants who do not understand multi-game theory, incentives can provide motivation to engage in barrier removal, fostering collaboration.
By cultivating the ability to think about and act upon barrier removal, it becomes possible to maintain and enhance cooperative structures.
The Superiority of Horizontal Segmentation
This is the overall strategy for rational decision-making in multi-games.
The strategy, derived from universal principles under the assumption of zero-knowledge multi-games, is applicable to any multi-game scenario.
By thinking about the entire multi-game system rather than focusing on the specifics of individual games, powerful strategies can be devised.
When analyzing complex phenomena, simplifying the subject while preserving its nature is crucial.
A common approach involves dividing the subject into finer units for analysis and then integrating the results. This can be referred to as vertical segmentation.
However, dividing a multi-game into individual games eliminates the nature of the multi-game itself, making vertical segmentation unsuitable.
An alternative is horizontal segmentation, which involves dividing a multi-game system based on the level of detail considered in each game.
In horizontal segmentation, analysis begins at the highest level, which contains the least detailed information. In multi-games, this corresponds to zero-knowledge games.
By analyzing based on common properties regardless of the specifics of individual games, universal strategies applicable to all scenarios can be developed.
Ownership Limits and Insurance Limits
In multi-game theory, analyzing limits using horizontal segmentation deepens universal strategies.
Notable examples include ownership limits and insurance limits.
Due to uncertainty, participants may suffer significant losses even with the most rational decisions. Such losses could render them unable to recover or force them out of the game entirely.
Thus, preparing for potential losses is crucial for each participant.
This preparation falls into two broad categories: ownership and insurance. Both have limits on the extent of losses they can absorb. However, as a general rule, insurance provides a greater capacity to withstand larger losses.
Whether it pertains to resources or abilities, the quantity or quality that one participant can cover through ownership is often less than what can be covered collectively through insurance.
Consequently, excessively increasing ownership to raise ownership limits at the expense of reducing the overall insurance limit results in weaker collective preparation for losses.
The Limits of Domination
Another critical concept is the limit of domination. While controlling other participants can provide an advantage in games, there is a limit to the number of participants one can dominate consistently.
In scenarios with many participants, it is challenging to maintain control over all of them indefinitely. Additionally, other participants with domination capabilities are likely to emerge, increasing the risk of rebellion from those under control.
Recognizing these limits reveals that pursuing unlimited ownership or domination is not a sound strategy in multi-games with many participants.
The aforementioned concepts of ownership and domination limits reinforce the conclusion that cooperation is rational in open-ended multi-games unless there is a specific reason to the contrary.
Ethical and Rationality Limits
Ethical appeals for cooperation among participants have inherent limitations. Therefore, in multi-games with many participants, it is essential to appeal to rationality, enabling each participant to build a cooperative framework for their self-interest.
Additionally, there is a limit to the complexity participants can rationally process.
Multi-game theory aims to reduce complexity through horizontal segmentation, thereby enhancing accessibility and enabling more participants to make appropriate rational decisions.
The theory also proposes designing incentives directly tied to tangible benefits, fostering a structure where more individuals can actively engage in strengthening cooperative systems.
Technological Limits
Advances in technology — whether scientific innovations or methods of control over others — enable greater impacts with smaller efforts or fewer resources.
This capability allows for actions that can cause significant harm to participants, executed by a small group with minimal resources.
While technology can also help mitigate or prevent harm, it is generally easier to cause harm than to prevent or reduce it.
As technology develops further, a situation may arise where a group of participants, willing to abandon the game themselves, attempts to cause the collapse of the entire system, including all participants. In such cases, other participants may find it impossible to prevent this outcome.
Therefore, technological progress must have limits. To prevent exceeding these limits, it becomes necessary to halt the race for technological advancement at some point.
This makes constructing a cooperative framework among participants not only rational but also essential for the survival of the multi-game itself. Without curbing the technological race before it exceeds these limits, the collapse of the entire system becomes inevitable.
Conclusion
By considering open-ended multi-games rather than relying solely on conventional game theory approaches, rational strategies diverge significantly.
There is a common misconception that incorporating multiple games and viewing the system as a whole increases complexity, making rational conclusions harder to reach. However, as demonstrated here, horizontal segmentation can effectively reduce complexity and derive rational strategies.
Conversely, prioritizing vertical segmentation of individual games while neglecting the multi-game perspective risks significantly suboptimal decisions, given the same cognitive capacity, time, and resources. This is highly irrational.
Furthermore, the choices we make and the outcomes of our decisions are recorded in our minds and societies. These records can either encourage cooperative choices in future decisions or do the opposite. In open-ended multi-games, achieving cooperation ultimately provides significant benefits and insurance. Therefore, unless there is a specific reason otherwise, increasing cooperative tendencies within this accumulation is rational.
Expanding this cooperative record collectively across society and maximizing its accumulation forms the path forward. This is not about morality or an ideal society; it is a narrative where cooperation becomes natural and rational due to the self-interest of each individual and the societies to which they belong.
This is a re-writing of the competitive narrative, shaped by the flawed perspective of vertical segmentation, into a cooperative narrative based on logical reasoning from the perspective of horizontal segmentation.
