I. Introduction
In the international open large-scale competition system, internationally licensed gambling groups play an important role in setting game rules, which have a significant impact on the capital of the entire sports industry. For every major global event, such as the World Cup, gambling companies provide odds for all participating teams, and fans around the world will place their bets based on their preferences. [1]
The setting of these odds involves very complex mathematical analysis and is the core of the entire competitive game. Because the odds are 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 teams, they are subjectively determined by the gambling companies. The ideal situation for gambling companies is that the results of any match can offset the gains and losses of the players' chips, allowing the gambling companies to earn risk-free fees. This is a very ideal and completely normal business model.
However, due to the inherent uncertainty in competitive sports and the natural bias of fans, in some cases, especially in important matches that attract global attention, a large amount of betting may be placed in one direction. This can lead to a situation where if the underdog wins, the gambling group will make excessive profits, and a small number of winning players will also receive huge profits. However, if the majority of players win, the gambling group will face huge payouts.
Although today's odds system has developed into very complex mathematical models and has implemented a dynamic mechanism to adjust odds in real time through the internet, sometimes fans' strong preferences for certain teams can significantly affect the true strength. In many extreme cases, this can lead to risks for gambling groups. For example, in the 2014 World Cup semi-final match between Germany and Brazil, the two teams were closely ranked and had similar levels. In theory, the odds should not differ much, but Brazil had the home advantage, and the 2014 Brazilian team was shining with stars. Thanks to the rapid development of the internet globally, the Brazilian team had a massive number of fans, leading to a historically rare one-sided betting situation. The vast majority of chips were placed on Brazil to win and advance to the final, putting the gambling companies in a dilemma of making huge profits or losses. They were forced to become the counterparty to the vast majority of funds, which is unacceptable for any gambling group. Although there is no evidence of match-fixing, in this match, the German team won against the biggest favorite, Brazil, with a score of 7:1 on Brazilian soil, a result that was almost unimaginable before the match, and hardly any players guessed it correctly. From the result, the gambling companies were the biggest beneficiaries. In all international events, fans have summarized a rule without scientific basis, "the big favorite is doomed to lose." However, this rule is not in line with probability theory and indirectly proves the existence of information asymmetry affecting match results.
Traditional gambling groups, although not aiming to participate in gambling for profit, have a probability of having to pay out more bets. To prevent human intervention in matches from the source, it is not about strictly enforcing laws and regulations to eliminate human intervention, but rather about changing the traditional game mechanism. With the increasing maturity of blockchain technology, the transparency, decentralization, and programmability of blockchain technology can make game rules unchangeable by anyone. Through the combination of multiple standard protocols, this paper proposes a new game contract protocol, CP505, based on mean field game theory.
II. Related Work
2.1 Mean Field Games (MFG)
The mean field game theory proposed by Pierre-Louis Lions and others in 2006-2007 provides equilibrium solutions for games involving a large number of homogeneous agents. This theory mathematically describes how individuals make optimal decisions based on the statistical behavior of other participants in a system with a large number of participants.
2.2 Game Theory
Game theory 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 betting games.
2.3 Market Mechanism Design
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, which forms the technical basis for creating transparent and tamper-proof gambling platforms.
2.5 Behavioral Economics
Behavioral economics combines psychology and economics to study the irrational behavior of people in economic decision-making, which is important for understanding and designing user interactions in gambling games.
2.6 Tournament Betting Market Analysis
Analysis of tournament betting markets, including odds setting, market liquidity, and information efficiency, provides an empirical research basis for designing gambling games.
2.7 Prisoner's Dilemma
A classic two-player non-cooperative game model in which the decision-making based on individual optimal choices leads to a suboptimal result for all participants. This concept was first proposed by Albert W. Tucker in 1950.
2.8 Computational Complexity of Multiplayer Games
As the number of game participants increases, finding equilibrium solutions becomes significantly more difficult. This is because the strategy space of the game grows exponentially with the number of participants, making it more complex to compute equilibrium.
2.9 Equilibrium of Multiplayer Games
In multiplayer games, Nash equilibrium may not exist or may be difficult to find, because each participant's optimal response strategy depends on the strategies of all other participants, and each person's strategy space is large.
III. Theoretical Basis and Model Construction
3.1 Application of Mean Field Game Theory in Assumptions
If each bet from users can be fragmented into countless pieces for trading, and the market can freely price these pieces, which can then be freely used for new bets, this transforms the traditional odds into a financial method. The problem of analyzing user bets is transformed into analyzing user financial behavior, and then into a game strategy problem with nearly infinite homogeneous opponents.
In classical game theory, games occur between opponents in the scene and usually only involve two people, such as the famous prisoner's dilemma. Games involving three opponents are very difficult to compute and reach equilibrium, which is why the Western film "The Good, the Bad and the Ugly" is so classic. If the number of players in the game reaches four, five, or more, it is mathematically unsolvable, meaning there is no so-called best strategy, and the game participants cannot adopt convergent strategies.
However, if the number of opponents in the game can be considered infinite, it is solvable mathematically. French mathematician and Fields Medalist Pierre-Louis Lions and several other mathematicians proposed the mean field game theory in 2006-2007, which can obtain the probability distribution at the equilibrium point for a game involving nearly infinite homogeneous opponents, and thus obtain the best strategy for game participants at the equilibrium point.
When mean field game theory was first proposed, people did not believe that this theory had any application in the financial field. The premise of establishing mean field game theory is that the opponents in the game are homogeneous. However, in traditional financial markets, the abilities and types of game opponents are completely different, including company management with insider knowledge and actual execution power, institutions and large accounts, and many individual investors. Because the game opponents are not homogeneous, manipulation always exists. For example, stock prices are not the result of a fair game. Large shareholders or management with insider information, or large funds that see the distribution of chips, are usually the manipulators of stock prices.
3.2 Mean Field Game Theory
Mean field game (MFG) theory specifically explores the strategies used by a large number of intelligent agents in a competitive environment, where each agent adapts to the actions of other agents around them to maximize their own benefits.
III. Theoretical Basis and Model Construction
3.1 Assumptions of Intelligent Agents
The assumptions of intelligent agents typically include the following:
1. Homogeneity: All intelligent agents are homogeneous, meaning they have the same preferences and decision-making abilities.
2. Large Number of Agents: There is a large number of intelligent agents in the system, to the extent that the behavior of individual agents has a negligible impact on the entire system.
3. Simplified Interaction: The interaction between intelligent agents is simplified to represent the average effect of agent behavior (i.e., mean field), rather than direct interaction between individuals.
4. Continuous Time: The behavior and decision-making processes of intelligent agents are typically modeled in a continuous time framework.
5. Rationality: Intelligent agents are assumed to be rational, meaning they will choose the optimal strategy to maximize their own interests.
6. Information Structure: In some models, intelligent agents may have different information structures, such as complete or incomplete information.
7. Strategy Selection: Intelligent agents adjust their strategies based on the average behavior of other intelligent agents to maximize individual utility.
8. Stability and Equilibrium: The behavior of intelligent agents tends towards a certain equilibrium state, such as Nash equilibrium, which is a key focus of MFG theory analysis.
9. Distributed Decision-Making: The decision-making process of intelligent agents is distributed, without a central coordinating authority.
3.3 Building Similar Assumptions of Intelligent Agents
In traditional odds-making, because the odds are set by gambling companies, all fan bets are based solely on their level of affection for a team or objective estimation, and whether there is arbitrage space in the odds set by the gambling company. Most users' individual behavior cannot influence 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, betting users cannot withdraw their bets or change their strategies. Once the bet is placed, there is no opportunity for regret. This does not conform to the assumptions of mean field game theory.
However, when applying blockchain technology and smart contract technology, allowing each user to fragment their bets and create highly liquid tradable assets, with market users determining the prices of these fragments, it indirectly allows users to change their strategies and influence others' strategies. 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 make a large number of participating users approximate intelligent agents, then according to mean field game theory, it is possible to have an optimal solution. This optimal solution is often a complex combination of Nash equilibria.
3.4 Overview of Nash Equilibrium Characteristics
1. Non-Cooperative: In non-cooperative games, each intelligent agent independently chooses its optimal strategy without considering the interests of other intelligent agents.
2. Strategy Combination: Nash equilibrium is a specific combination of strategies for all intelligent agents. In the equilibrium state, each intelligent agent's strategy is the best response to the strategies of other intelligent agents.
3. Stability: Nash equilibrium is a stable state, meaning that without external intervention, no intelligent agent will benefit from changing its strategy.
4. Predictive: In game theory, Nash equilibrium provides a method for predicting game results 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 result.
6. Rationality Assumption: The existence of Nash equilibrium is based on the assumption that intelligent agents are rational, meaning they will choose strategies to maximize their own interests.
7. Utility Maximization: In the equilibrium state, each intelligent agent chooses a strategy that maximizes its own utility given the strategies of other intelligent agents.
3.5 Theoretical Framework of the Assumed Model
In a gambling game with a large number of players and no central group capable of intervening in the rules, these large numbers of players are homogeneous intelligent agents, meeting the conditions for the establishment of mean field game theory. At the same time, these players cannot reach a cooperative game with a large number of other players, so mean field game theory also belongs to non-cooperative games.
Nash equilibrium brings an important value to us, which is that in this model, all users are no longer engaged in "gambling," because under non-cooperative conditions, if they are rational, they can only adopt a certain strategy, or a dominant strategy, which is most advantageous to themselves. Nash equilibrium is usually effective for a small number of players, where rational players adopt dominant strategies, achieving a certain equilibrium. Both mean field game theory and Nash equilibrium are based on non-cooperative games. The equilibrium reached by mean field game theory can be understood as a combination of countless Nash equilibria.
Traditional odds gambling can only be a zero-sum game under given odds. Once the largest participant (the gambling group) discovers the risk of a huge payout, it is highly likely to intervene in the game results in various ways, leading to significant unfairness. However, in the new game model under the CP505 protocol, users have the opportunity to choose their own strategies and can achieve multiple strategies. Each decision made will affect others, and countless intelligent agents ultimately have the opportunity to achieve Nash equilibrium and reach the optimal solution. This optimal solution does not mean that all users will profit, but under the premise of fairness and transparency, all users have made rational decisions and achieved their strategies independently. This is a completely new game design, no longer traditional "gambling."
In a tournament format, after the results of each round are determined, all players receive the same information about the changed conditions. Players adjust their strategies and execute them based on the changed conditions and by observing the behavior of other players. After each round of results is determined, using the mathematical formulas of mean field game theory, the theoretical equilibrium value can be calculated based on the probability distribution of the survival probabilities of each team and the odds of each team becoming the final winner generated by the free trading of players. This equilibrium value represents a pricing of a series of teams and chips. Emotions of players may cause actual pricing to deviate from theoretical pricing, and rational traders (arbitrageurs) will trade this deviation, causing actual pricing to tend towards theoretical pricing. The presence of both arbitrageurs and traders with emotional preferences in the market will generate sufficient trading activity, benefiting market liquidity.
3.6 Assumptions of the Game Model Based on the CP505 Protocol
Based on the above analysis, the game model design under the CP505 protocol should fully consider the following assumptions:
1. All match information is publicly transparent.
2. All game rules cannot be tampered with by anyone.
3. Even if there are differences in game results, they will not affect game strategies.
4. No centralized group has the ability to intervene in any rule setting. Even if a game is intervened, it has no effect on collective strategies.
5. Each participant is homogeneous, pursuing the highest return rather than "odds," and can adjust their behavior based on the strategies of other participants.
6. The impact of a single intelligent agent's behavior on the entire system is negligible.
7. Market prices are determined by sufficient market competition and liquidity, and the dynamic changes in market prices, representing the state of all intelligent agents and the probability distribution of strategies in the market. This market pricing is considered an equilibrium result generated by mean field game theory.
3.7 Technical Safeguards of the Model Based on Blockchain Technology and Smart Contracts
Blockchain technology and Ethereum-based smart contract technology can ensure that all data is publicly accessible and traceable. Using a decentralized, distributed ledger network, programs can be recorded on all network nodes, and no one has the ability to tamper with established rules.
3.8 Model Construction
1. Convert all bets on participating teams into NFT assets based on the ERC721 protocol. These assets can also be traded in a decentralized manner.
2. Users purchasing an NFT for any team represent a specific type of bet.
3. All bets are not controlled by any centralized group, but are held by smart contracts and distributed to the ultimate winner by the smart contract.
4. Based on the CP505 protocol, all NFTs can be destroyed and converted into ERC20 universal tokens. However, each time an NFT is destroyed, a portion of the ERC20 tokens obtained will be permanently destroyed by entering a black hole address.
5. The token is traded in a decentralized trading market based on an Automated Market Maker (AMM) model to avoid any human intervention.
6. A certain amount of ERC20 tokens can be used to re-synthesize an NFT card for a specific team, meaning a new bet. This can generally be randomly generated. If a user is not satisfied with the randomly generated team, they can destroy the NFT again, obtain tokens, and generate again.
7. Each user's actions of destroying and synthesizing will lead to continuous token destruction, thereby affecting the token's price in the secondary market. Buyers in this market need to purchase tokens to synthesize new team cards, while sellers need to reduce losses through token sales, or even reduce their own risks by buying low and selling high. The market price will be a continuous equilibrium price formed by mean field game theory. The repeated, free, and rational process of destroying and generating NFTs by users is a full expression of individual free choice of strategy.
8. After the final match, all holders of the champion team's NFT cards will share the total bet amount in the contract. In theory, every user has sufficient time after the final match to synthesize the champion's card.
9. The final result achieved by this model is a series of equilibrium prices generated by mean field game theory.
IV. CP505 Business Design Plan
Assumption: There are 36 teams competing for the championship in a large-scale tournament. The tournament lasts for 1 month. It is known which 36 teams are participating, and the tournament results are a physical world public event with unique determinism. Theoretically, this assumption can be applied to any competitive event.
The first NFT blind box. Each blind box randomly generates bets for five teams. Each bet is completely identical. For example, if a blind box costs $100, then the five team NFTs randomly drawn from it are valued at $20 each. This $20 can be considered as a bet.
NFT trading market: The trading price of popular team NFTs will rise until it reaches an equilibrium price. The price of unpopular teams will theoretically drop significantly due to lack of purchasing demand. This is the first market equilibrium game.
According to the CP505 protocol mechanism, NFTs can be destroyed and produce a fixed amount of ERC20 tokens - V-Tokens, which can then be used to re-synthesize blind boxes. This allows users the opportunity to obtain NFT chips for teams they are relatively satisfied with.
The V-Tokens generated from destroying NFTs are controlled by smart contracts. 10% of the V-Tokens are sold on the decentralized trading market and added to the total prize pool, increasing the total prize money. Another 10% of the V-Tokens are sent to a black hole address for permanent destruction.
The holder of the NFT for the eventual champion team shares the prize pool.
Player Strategy Considerations
For participants, the actions they can take include but are not limited to the following strategies:
Purchase a large number of blind boxes to obtain NFTs for popular teams, eliminate NFTs for less valuable teams, synthesize new blind boxes, and gradually turn their team NFTs into championship cards to win prizes.
Sell NFTs for popular teams whose prices have been inflated, buy NFTs for teams they favor, and gain investment returns from NFTs.
Fragment NFTs for teams they do not favor, generate V-Tokens, and choose to sell them to recover some costs, or use V-Tokens to re-synthesize blind boxes to continue pursuing the randomness of the game.
As the group stage or knockout stage progresses, the value of each team's NFT will change, driven by the randomness of match results. As the value of team NFTs changes, it will drive participants to take appropriate actions, such as buying/selling team NFTs or fragmenting NFTs/synthesizing blind boxes.
Players can also observe the price of V-Tokens. With an increase in the number of eliminated teams leading to an increase in fragmentation, the price of V-Tokens may fall below the theoretical price due to insufficient purchasing volume. Players purchasing V-Tokens to synthesize new NFTs can generate additional profits. Similarly, if an increase in the value of the total prize pool leads to speculative demand for purchasing V-Tokens, the price of V-Tokens may exceed the theoretical value. At this point, selling V-Tokens generated from fragmenting NFTs for teams that have not been eliminated but have little hope of winning the championship may be profitable.
V. Open Source Smart Contract Code
https://github.com/ai77simon/cp505/
The development of this code received partial support from the independent business team Euro505 in Singapore. They conducted a social experiment based on the European Cup according to this paper, and the experimental data will be further presented to readers in the next paper.
VI. Conclusion
The CP505 protocol, built on blockchain technology, has pioneered a new game concept for all tournament formats. Its theoretical basis comes from mean field game theory, Nash equilibrium, behavioral economics, and other theories. Technically, it can only be achieved through complete decentralization, public transparency, and tamper-proof blockchain technology, as well as various decentralized NFT trading markets and decentralized token trading markets. In this game involving an infinite number of homogeneous individuals, all information is publicly transparent, and users can repeatedly modify their strategies, thereby influencing the strategies of others. Ultimately, under any temporary equilibrium state (before the randomness of the next match result is determined), theoretically all users collectively determine the optimal strategy. The direct embodiment 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 prices generated by trading may deviate from the theoretical equilibrium price. Rational arbitrageurs will trade this price deviation, buying low and selling high, ultimately driving the trading price towards the theoretical price. All prices are generated by the emotional preferences of players in the market and rational arbitrageurs through trading, not manipulated or created behind the scenes. The different purposes and strategies of arbitrage players and players pursuing personal preferences for participating teams will increase market activity, making the market healthier.
In another sense, this rule design, under technological innovation, is an attempt by humans to use mathematical game theory to break the traditional odds-based gambling mechanism and achieve a new game enjoyment focused on investment rather than gambling.
Due to the limited abilities of the authors, there are shortcomings in all design thinking and development work. We hope that this research can inspire more scholars and are willing to accept criticism and corrections from any scholars.
References
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