平均場ゲーム理論に基づくカップ競争システムにおけるCP505プロトコルのゲームメカニズムの革新

I.はじめに

In the internationally open large-scale cup competition system, international legal betting groups play an important role in formulating the rules of the game, which has a significant impact on the entire sports industry capital. For every major world event, such as the World Cup, betting companies give odds for all participating teams, and fans around the world will choose their own teams to bet on according to their preferences. [1]

The odds setting involves very complex mathematical analysis and is the core of the entire competitive game. The odds are weighted and calculated based on a series of indicators such as the strength of the participating teams, the current status of the players, and the historical performance of the team, and are given subjectively by the bookmaker. The ideal situation for the bookmaker is that the results of any game can offset each other, and the bookmaker earns risk-free commissions. This is a very ideal and completely normal business model.

However, because there are many contingencies in competitive sports and fans have a natural tendency, at certain times, when it comes to important matches that attract global attention and huge amounts of global bets, a large number of bets will be placed in a single direction. As a result, if the match turns out to be an upset and most players guess wrong, the betting group will have excess profits, and a small number of players who guess correctly will also receive huge profits, but if most players guess correctly, the betting group will face huge compensation.

Although todays odds system has developed into a very complex mathematical model and a dynamic mechanism for real-time adjustment of odds through the Internet, sometimes fans love for certain teams will seriously affect the actual strength of the teams. Many extreme situations will cause gambling groups to face risks. For example, in the 2014 World Cup semi-finals between Germany and Brazil, the two teams were ranked and level close, and theoretically the odds should not be much different, but Brazil had a home advantage, and the Brazilian team in 2014 was shining. Thanks to the rapid development of the Internet around the world, the Brazilian team had a large number of fans, which led to a rare one-sided bet at that time. Most of the chips were placed on Brazils final victory and advancement to the finals. Bookmakers faced a dilemma of making a big profit or a big loss, and were forced to become the opponent of most funds, which was unacceptable to any gambling group. Although there is no evidence that the game was manipulated, in this historical match, the German team defeated the most popular Brazilian team with a home advantage with a score of 7:1 at Brazils home court and won and advanced. This score that was unimaginable before the game was almost not guessed by players. Judging from the results, bookmakers are the biggest beneficiaries. In all international competitions, fans have concluded a rule that has no scientific basis: the favorites must die. But in fact, this is due to the huge risks brought about by zero-sum games. Letting the favorite team die is the most helpless way to reduce business risks. This simple rule, which is so inconsistent with probability theory, indirectly proves that there is information asymmetry that interferes with the results of the game.

Although traditional gambling groups do not aim to participate in gambling in terms of business model, the simple odds betting method will definitely require gambling groups to pay out more bets. If you want to curb human intervention in the game from the source, it is definitely not to formulate laws and regulations and strictly enforce them to eliminate human intervention, but to change the traditional game method where the dealer actively gives odds. With the increasing maturity of blockchain technology, the transparency, decentralization and programmability of blockchain technology can make the rules of the game impossible to be tampered with by anyone. Through the combination of multiple standard protocols, this paper proposes a new game contract CP 505 protocol based on mean field game theory.

II. Related Work

2.1 Mean Field Games (MFG)

The mean field game theory [2] proposed by Pierre-Louis Lions et al. in 2006-2007 provides an equilibrium solution for games involving a large number of homogeneous intelligent agents. The theory mathematically describes how individuals in a system with a large number of participants make optimal decisions based on the statistical behavior of other participants.

2.2 Game Theory

Game theory[3] is a mathematical theory that studies the interaction between decision makers with conflicting and cooperative characteristics. It provides a framework for understanding and predicting strategic behavior in tournament-based gambling games.

2.3 Market Mechanism Design [4]

Market mechanism design focuses on how to design market rules to achieve specific economic goals, such as efficiency, fairness, and transparency.

2.4 Cryptocurrency and Blockchain Technology

Cryptocurrency and blockchain technology provide a decentralized value transfer mechanism that provides the technical foundation for creating transparent and immutable gaming platforms. [5]

2.5 Behavioral Economics

Behavioral economics combines psychology and economics to study people鈥檚 irrational behavior in economic decision-making, which is important for understanding and designing user interactions in gambling games. [6]

2.6 Tournament Betting Market Analysis

The analysis of the tournament betting market, including odds setting, market liquidity and information efficiency, provides an empirical research basis for designing betting games. [7]

2.7 Prisoner鈥檚 Dilemma

A classic two-player non-cooperative game model in which each players decision, based on his or her individual best options, leads to a poor outcome for all players. The concept was first proposed by Albert W. Tucker in 1950. [8]

2.8 Computational Difficulty of Multiplayer Games

As the number of game players increases, the difficulty of finding an equilibrium solution increases significantly. This is because the strategy space of the game grows exponentially with the number of players, making it more complicated to calculate the equilibrium. [9]

2.9 Equilibrium of Multiplayer Games

In multiplayer games, Nash equilibrium may not exist or be difficult to find because the optimal response strategy of each player depends on the strategies of all other players, and the strategy choice space of each player is very large. [10]

3. Theoretical basis and model construction

3.1 Application of mean field game theory in hypothesis

If each bet of the user can be turned into countless fragments for trading, and the market can freely price the fragments, and these fragments can freely realize new bets, this will transform the traditional odds method into a financial method. The problem will be transformed from analyzing and studying the users betting problem to analyzing the users financial behavior, and then into a game strategy problem with nearly infinite homogeneous opponents.

In classic game theory, games take place between opponents in a scenario, usually involving only two people, such as the famous prisoners dilemma. Games involving three opponents are computationally very difficult and it is difficult to reach equilibrium, which is why the western The Good, the Bad and the Ugly is so classic. If the number of participants in the game reaches four, five or more, it is mathematically unsolvable. What is unsolvable here means that there is no so-called optimal strategy, so the participants in the game cannot adopt convergent strategies.

However, if the number of opponents in the game can be considered infinite, there is a mathematical solution. French mathematician and Fields Medal winner Pierre-Louis Lions and several other mathematicians proposed the mean field game theory from 2006 to 2007. For a game with nearly infinite homogeneous opponents, the probability distribution in the equilibrium state can be obtained mathematically, thereby obtaining the best strategy of the game participants at the equilibrium point.

When the mean field game theory was first proposed, people did not think that this theory had any application in the financial field. The premise of establishing the mean field game theory is that the opponents of the game are homogeneous. In the traditional financial market, the abilities and types of the opponents are completely different. There are company management with insider knowledge and actual execution, institutions and large accounts, and many individual investors. Because the opponents of the game are different, there is always manipulation. For example, the stock price is not the result of a fair game. The major shareholders or management who have insider information, or the big funds who have seen the distribution of chips, are usually the manipulators of the stock price.

3.2 Mean Field Game Theory

Mean field game (MFG) theory specifically explores the strategies used by a large number of agents in a competitive environment. Each agent will respond to the actions taken by other agents around it in order to maximize its own benefits.

The assumptions of the agent usually include the following:

• 1. Homogeneity: All agents are homogeneous, i.e. they have the same preferences and decision-making abilities.

• 2. A large number of agents: There are so many agents in the system that the behavior of a single agent has a negligible impact on the entire system.

• 3. Simplification of interactions: The interactions between agents are simplified by the average effect of the agent鈥檚 behavior (i.e., the mean field), rather than by direct interactions between individuals.

• 4. Continuous time: The behavior and decision-making process of intelligent agents are usually modeled in a continuous time framework.

• 5. Rationality: Agents are assumed to be rational, that is, they will choose the optimal strategy based on their own goal of maximizing their own interests.

• 6. Information structure: In some models, agents may have different information structures, such as complete information or incomplete information.

• 7. Strategy selection: The agent will adjust its strategy based on the average behavior of other agents to maximize individual utility.

• 8. Stability and equilibrium: The behavior of intelligent agents will tend towards a certain equilibrium state, such as Nash equilibrium, which is one of the focuses of MFG theoretical analysis.

• 9. Distributed decision making: The decision making process of the agent is distributed and there is no central coordinating body.

3.3 Constructing Similar Agent Hypotheses

In the traditional odds system, since the odds are set by the bookmaker, all fans bet only based on their own preference for the team or objective estimation, and whether there is arbitrage space in the odds set by the bookmaker. The personal behavior of most users cannot affect the behavior of others, and the betting behavior of others will not affect my betting behavior. When the odds change due to the behavior of a large number of users, the betting users cannot withdraw their bets or change their strategies. Once they make a decision, there is no chance to regret it. This does not conform to the assumption of mean field game.

However, when blockchain technology and smart contract technology are applied, each user is allowed to fragment his or her bets to form highly liquid trading products. The fragment prices are determined by market users for the second time, which indirectly enables users to change their strategies and thus influence the strategies of others. The behavior of these users is very close to the behavior of intelligent agents in mean field game theory.

Once our model has the opportunity to enable a large number of participating users to become approximate intelligent agents, then according to the mean field game theory, it is possible that an optimal solution will emerge, and this optimal solution is often a set of complex Nash equilibrium.

3.4 Overview of Nash Equilibrium Characteristics

• 1. Non-cooperation: In a non-cooperative game, each agent independently chooses its own optimal strategy without considering the interests of other agents.

• 2. Strategy combination: Nash equilibrium is a specific combination of all agent strategies. In equilibrium, each agent鈥檚 strategy is the best response to the strategies of other agents.

• 3. Stability: Nash equilibrium is a stable state, that is, in the absence of external intervention, no agent will benefit from changing its strategy.

• 4. Predictability: In game theory, Nash equilibrium provides a way to predict the outcome of a game because it represents a self-reinforcing strategy state.

• 5. Possible multiple equilibria: In some games, there may be multiple Nash equilibria, each representing a possible game outcome.

• 6. Rational assumption: The establishment of Nash equilibrium is based on the fact that intelligent agents are rational, that is, they will choose strategies based on their own goal of maximizing their own interests.

• 7. Utility maximization: In equilibrium, each agent chooses a strategy that maximizes its own utility given the strategies of other agents.

3.5 Theoretical framework of the hypothesis model

In gambling games with a large number of players, in the absence of a dealer, these large numbers of players are homogeneous agents, which meets the conditions for the establishment of mean field games. At the same time, these players cannot reach a cooperative game with a large number of other players, so mean field games are also non-cooperative games.

Nash equilibrium brings us an important value, that is, all users in this model are no longer gambling, because under non-cooperative conditions, if the user is rational, he can only adopt a certain strategy, or a dominant strategy, which is the most beneficial to him. Nash equilibrium is usually effective for a small number of players. Rational players all adopt a dominant strategy and reach a certain equilibrium. The premise of mean field game and Nash equilibrium is non-cooperative game. The equilibrium achieved by mean field game can be understood as the combination result of countless Nash equilibria.

Traditional odds gambling can only be a zero-sum game under given odds. Once the largest participant (gambling group) finds that there is a huge risk of compensation, it is very likely to intervene in the results of the game in various ways, resulting in great unfairness. The new game model under the CP 505 protocol has the opportunity to let users choose their own strategies and implement multiple strategies. Every decision will affect others, and countless intelligent agents will eventually have the opportunity to achieve Nash equilibrium and achieve the optimal solution. This optimal solution is not to make all users profitable, but under the premise of fairness and transparency, all users have fully and autonomously implemented their own strategies based on their own rational decisions. This is a new game design, not the traditional gambling.

In a cup competition system, after the results of each round of games are announced, all players receive the same information about the change in conditions. Players redefine their strategies and execute them based on the changes in conditions and the behavior of other players. After the results of each round are determined, the theoretical equilibrium value can be calculated based on the probability of each teams continued survival and the odds of each team becoming the final winner generated by the free trade of players, using the theoretical mathematical formula of mean field game. This equilibrium value is the pricing of a series of teams and chips. Players emotions may cause the actual pricing to deviate from the theoretical pricing, and rational traders (arbitrageurs) will trade this deviation, making the actual pricing tend to the theoretical pricing. The existence of arbitrageurs and traders with emotional preferences in a market will allow the market to generate enough transactions, which is beneficial to the markets activity.

3.6 Assumptions of the game model based on the CP 505 protocol

Based on the above analysis, the design of the game model of the CP 505 protocol should fully consider the following assumptions:

• 1. All competition information is open and transparent

• 2. All game rules cannot be tampered with by anyone

• 3. Even if the game results are different, it will not affect the game strategy

• 4. No centralized group has the ability to interfere with any rule setting. Even if it interferes with the game, it will have no impact on the collective strategy.

• 5. Each participant is homogeneous. They all pursue the highest rate of return rather than the odds. They can repeatedly adjust their behavior based on the strategies of other participants.

• 6. The impact of a single agent鈥檚 behavior on the entire system is negligible.

• 7. The market price is determined by sufficient market competition and liquidity. The market price is dynamically changed by all participants through repeated games. Its changes reflect the state and probability distribution of strategies of all intelligent entities in the market. The market pricing is regarded as an equilibrium result produced by a mean field game.

3.7 Blockchain technology and smart contracts provide technical support for the model

Blockchain technology and Ethereum-based smart contract technology can make all data publicly searchable and traceable. By utilizing a decentralized, distributed accounting network, programs can be recorded on all network nodes, and no one can tamper with the established rules.

3.8 Building the Model

• 1. Convert all bets of participating teams into NFT assets based on the ERC 721 protocol. This asset can also be traded in a decentralized manner.

• 2. When a user purchases the NFT of any team, it represents a special type of bet.

• 3. All bets are not controlled by any centralized group, but are kept by smart contracts and distributed to the final winners by smart contracts.

• 4. Based on the settings of the CP 505 protocol, all NFTs can be destroyed and converted into ERC 20 universal tokens. However, a portion of the ERC 20 tokens obtained from each destruction of NFTs will be permanently destroyed in the black hole address.

• 5. The token is traded on a decentralized exchange based on the automated market maker (AMM) model, avoiding any human intervention.

• 6. A certain number of ERC 20 tokens can be used to re-synthesize a team鈥檚 NFT card, which means re-betting. Generally, it can be randomly generated. If the user is not satisfied with the randomly generated team, the NFT can be destroyed again, the token can be obtained, and it can be generated again.

• 7. Each users destruction and synthesis according to his own decision will lead to the continuous destruction of tokens, which will in turn affect the price of the token in the secondary market. Buyers in this market need to buy tokens to synthesize new team cards, and sellers of tokens need to reduce losses through the sale of tokens, or even reduce their own risks by buying low and selling high. The market price will be a price formed by a continuous mean field game. The repeated, free, and rational destruction and generation of NFTs by users is a full expression of the individuals free choice strategy.

• 8. After the finals, all users holding the champion team鈥檚 NFT cards will share all bets in the contract. In theory, every user will have enough time to synthesize the champion鈥檚 card after the finals.

• 9. The final result of this model is mathematically expressed as the equilibrium price generated by a series of mean-field games.

4. CP 505 Commercial Design Program

Assumption: There is a large-scale competition in the market with 36 teams competing for the championship. The competition lasts for 1 month. It is already fully known which 36 teams are, and the result of the competition is a public event in the physical world with unique certainty. In theory, any competitive game can realize this assumption.

The first NFT blind box. Each blind box randomly generates five teams bets. Each bet is exactly the same. For example, if a blind box is worth $100, then five teams NFTs will be randomly opened, and each NFT is worth $20. This $20 can be considered as a bet.

In the NFT trading market, the trading price of popular teams will rise until an equilibrium price is formed. Since there is no demand for unpopular teams, the price will theoretically fall a lot. This is the first market game equilibrium.

According to the mechanism of CP 505 agreement, NFT can be destroyed and a fixed ERC 20 token, V-Token, can be produced, and then V-Token can be used to re-synthesize blind boxes. The advantage of this is that users have the opportunity to obtain NFT chips of the team they are relatively satisfied with.

The V-Tokens generated by destroying NFTs are controlled by smart contracts, and 10% of the V-Tokens are sold on decentralized exchanges and added to the total prize pool. This increases the total user bonus. The other 10% of the V-Tokens are sent to a black hole address for destruction.

The holders of the final champion teams NFT share the prize pool.

Player strategy thinking

For participants, the actions they can take include but are not limited to the following strategies:

• Buy a large number of blind boxes to get popular team cards, eliminate worthless team cards, synthesize new blind boxes, and gradually turn the team cards in your hands into championship cards to get bonuses

• Sell the NFTs of popular teams whose prices have been inflated, buy the NFTs of the teams he is optimistic about, and obtain NFT investment returns.

• By breaking up the NFT of the team he is not optimistic about and generating V-Token, he can choose to sell it to recover some costs, or use the V-Token to re-synthesize the blind box to continue pursuing the contingency of the game.

• As the group stage or knockout stage progresses, the value of each teams NFT will change, and the driving factor for this value change comes from the randomness of the game results. As the value of the teams NFT changes, it will drive participants to take actions they think are appropriate, such as buying/selling team NFTs, or fragmenting NFTs/synthesizing blind boxes.

• Players can also observe the price of V-Token. As the number of eliminated teams increases, the fragmentation increases. The price of V-Token may be lower than the theoretical price due to insufficient purchases. Players who purchase V-Tokens to synthesize new NFTs will gain additional benefits. Similarly, if the increase in the value of the total prize pool causes an increase in speculative demand for V-Tokens, the price of V-Tokens will exceed the theoretical value. At this time, it may be profitable to sell the V-Tokens generated by the fragmentation of NFTs of teams that have not been eliminated but have little hope of winning the championship.

5. Open Source Smart Contract Code

https://github.com/ai77simon/cp505/

The writing of this code was partially supported by the euro 505 group, an independent business team in Singapore. They conducted a social experiment based on the European Cup based on this paper, and we will further show the experimental data to readers in the next paper.

VI. 結論

The CP 505 protocol built on blockchain technology has created a new game theory for all cup-based tournaments. Its theoretical basis comes from theories such as mean field games, Nash equilibrium, and behavioral economics. Technically, it must be fully decentralized, open, transparent, and tamper-proof blockchain technology, as well as the industrial cooperation of many decentralized NFT trading markets and decentralized token trading markets. In this game, which is equivalent to the participation of an infinite number of homogeneous individuals, all information is open and transparent, and users can modify strategies repeatedly, thereby affecting the strategies of others, and finally achieve any short-term equilibrium state (the randomness of the results of the next round of games has not yet been generated). In theory, all users jointly determine the optimal strategy, and the direct manifestation of this optimal strategy is the price (including the team NFT price and the V-Token price).

Because players always have various preferences and emotions, the price generated by the transaction may deviate from the theoretical equilibrium price. At this time, there will be rational arbitrageurs who trade this price deviation, selling high and buying low, so that the transaction price will eventually tend to the theoretical price. All prices are generated by emotional preference players and rational arbitrageurs in the market through transactions, not by manipulation or shady dealings. The different purposes and strategies of arbitrageurs and players who pursue personal preferences for participating teams will increase the activity of the market and make the market healthier.

In another sense, the design of this rule is an attempt by humans, under technological innovation, to use mathematical game theory to break the traditional odds-based gambling mechanism and achieve a new gaming experience that is not for the purpose of gambling but for investment.

Due to the limited ability of the authors, all design thinking and development work have shortcomings. I hope that this study can bring inspiration to more scholars, and I am willing to accept criticism and correction from any scholars.

参照

[1] Asch, P. (2017). Mathletics: How Gamblers, Managers, and Sports Enthusiasts Use

Mathematics in Twenty-First-Century Baseball. Princeton University Press.

[2] Lions, P.-L. (2007). Mean field games. In: J. Math. Sci., Vol. 177, No. 3, pp. 415-430.

[3] Nash, JF (1950). Equilibrium points in n-person games. In: Proc. Nat. Acad. Sci.

USA, Vol. 36, No. 1, pp. 48-49.

[4] Myerson, RB (1981). Optimal auction design. In: Mathematics of Operations

Research, Vol. 6, No. 1, pp. 58-73.

[5] Nakamoto, S. (2008). Bitcoin: A peer-to-peer electronic cash system.

In: Bitcoin.org.

[6] Kahneman, D., Tversky, A. (1979). Prospect theory: An analysis of decision under

risk. In: Econometrica, Vol. 47, No. 2, pp. 263-291.

[7] Forrest, D., Simmons, R. (2006). Betting markets: A survey. In: Journal of

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[8] Tucker, AW (1950). A Two-Person Dilemma. In: Psychometrika, Vol. 17, No. 2,

pp. 186-202.

[9] Leyton-Brown, K., Shoham, Y. (2008). Multiplayer Games. In: Essentials of Game

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This article is sourced from the internet: Innovation of the game mechanism of CP505 protocol in the cup competition system based on mean field game theory

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