Amir, R. (2005). Supermodularity and complementarity in economics: An elementary survey. Southern Economic Journal, 71(3), 636-660. PDF
This text provides an overview of the literature on ultra-modular optimization and games in economics from the perspective of potential users. The methodology provides a new approach for comparative statics and allows for a general analysis of games with strategic complementarity. Results are presented in a simplified yet rigorous manner for special cases with one-dimensional sets of parameters and behaviors, without reference to lattice theory, with emphasis on wide accessibility. Detailed applications are presented for well-known models of consumer behavior, monopoly price shifting, Bertrand and Cournot competition, strategic R&D, search, and matching. Where appropriate, useful tips for application and comparative comments are inserted.
The main points are as follows
- Ultramodular optimization is a new comparative statics methodology that determines how changes in parameters affect the endogenous variables of the optimization model.
- It requires only the assumptions that are important for the monotonicity of the optimal solution, and can eliminate extraneous assumptions such as smoothness, interior point solutions, concavity, etc. that are required by the classical method.
- Under the strategic complementarity of the game, the existence of equilibrium points and comparative statics are established.
- The complementarity condition with the order concept is also presented, and monotonic comparative statics are shown under weaker conditions.
- Applications to various areas of economics are discussed, including consumer behavior, monopolies, oligopolies, and R&D.
In conclusion, it is stated that this methodology greatly expands the scope of application of comparative statics analysis of economic models.
Many words I don’t understand, explained for college students.
Yes, I will explain it more clearly for college students studying economics.
In economics, we often analyze how a change in one variable (e.g., price) affects another variable (e.g., quantity demanded). This is called comparative statics analysis.
Traditional comparative statics analysis requires the assumption that the model is mathematically “clean” (e.g., the function is smooth and differentiable, the solution is an interior point solution, etc.). In real economic problems, however, such assumptions often do not hold.
The “ultramodular optimization” method presented in this paper allows one to perform comparative statics analysis without such awkward mathematical assumptions. Key to this method is the concept of “strategic complementarity.
In game theory, there is “strategic complementarity” when players’ strategies are complementary to each other. For example, if one company increases its advertising, other companies have an incentive to increase their advertising in response. In this case, there is strategic complementarity in the strategies of the firms.
With an ultra-modular optimization approach, the existence of an equilibrium point and how the equilibrium point changes with respect to changes in the parameters (comparative statics) are guaranteed in situations of strategic complementarity.
This approach can be applied to many areas of economics, including microeconomics, industrial organization theory, and macroeconomics. For example, it is useful for analyzing real economic problems, such as firms’ pricing behavior, R&D investment, and consumer choice behavior.
The argument of this paper is that problems that are difficult to analyze using conventional methods of comparative statics analysis can be analyzed using ultra-modular optimization methods. The mathematical details may be difficult to understand, but I hope you now understand the basic idea. The important point is that the development of analytical methods in economics will enable economic analysis that is more realistic.
I’ve heard the term “network resources”, what’s the connection?
Let me talk about the relationship between “network goods” and ultra-modular optimization.
A network good is a good whose value depends on the number of people using it. For example, telephones, social networking services, and online games are network goods. Using the phone as an example, the more people use the phone, the more people there are to call, and the more valuable the phone becomes.
This very feature of network goods is closely related to “strategic complementarity”. If one person uses the phone, there is an incentive for others to use the phone as well. In other words, there is strategic complementarity in user behavior.
The ultra-modular optimization approach can be applied to the analysis of such network goods. For example, consider the pricing problem for network goods. For a network good, the optimal strategy may be to reduce the price and increase the number of users, since the value of the good increases as the number of users increases. The method of ultra-modular optimization is useful in analyzing how this optimal price is determined.
The method can also be applied to analyze the diffusion process of network goods. The number of initial adopters plays an important role in determining whether a good will spread widely in society. Using the method of ultra-modular optimization, it is possible to analyze under what conditions a good will spread rapidly (i.e., whether the so-called “critical point” will be exceeded).
Thus, the analysis of network goods is one of the key applications of ultra-modular optimization. Since network goods play a very important role in the modern economy, it is of great significance for economics to develop methods for their economic analysis.
With that in mind, I’d like to explain Beyond Public and Private.
Yes, based on the content of this white paper, I will explain the relationship between “network goods” and “ultra-modular goods”.
The argument of this white paper is that technological advances have led many goods to take on the characteristics of “ultra-modular goods”. An ultramodular good is a good that is more efficient to provide to many people at the same time than to provide individually. In other words, they are goods for which economies of scale operate.
The traditional concept of “network goods” can be viewed as part of this hypermodularity. Network goods such as telephones and social networking services have the property that their value increases as the number of users increases, which is a type of hyper-modularity.
What this white paper argues, however, is that a broader range of goods, not just network goods, are becoming hyper-modular in nature. For example, public health, digital infrastructure, and scientific research are more efficient when shared by all and have the characteristics of hyper-modular goods.
The problem is that our current economic system is not suited for the provision of such ultra-modular goods. Capitalism is well suited for transactions based on private property rights, but not for the joint provision of ultra-modular goods. As a result, ultra-modular goods tend to be underprovided or privately monopolized.
This white paper argues that new collective decision-making mechanisms are needed for the provision and management of ultra-modular goods. It states that the public sector, private sector, and communities must work together to combine democratic decision-making, market mechanisms, and community governance.
In short, this white paper argues that while network goods are part of an ultra-modular good, the problem is broader and requires a transformation of the entire economic system. The proper provision and management of ultra-modular goods will become an increasingly important issue as technology advances.
In other words, after “easy-to-understand network effects” such as the telephone and the Internet attracted attention and the term network resources was coined, the concept has been expanding as “there are many more things with similar characteristics. There are those who continue to call the expanded concept “network goods” and those who newly call it “ultra-modular goods”.
Supermodular function - Wikipedia
I think the mathematical concept of supermodularity is a generalization of the characteristics of continuous functions. I would also like to explain the chapter “Supermodularity in economics and game theory” and
As you point out, mathematically, supermodular is a concept that generalizes the characteristics of continuous functions. For a continuous function f, at all points x, y, f(max(x, y)) + f(min(x, y)) ≥ f(x) + f(y) f is defined to be supermodular when where max(x, y) and min(x, y) are the maximum and minimum values for each component of x and y, respectively. Intuitively, a supermodular function has the property that the increase in the function value when the variables increase simultaneously is greater than the sum of the increases in the function values when the variables increase individually. In other words, it is a function in which there is a synergistic effect between the variables. Applications to economics and game theory are discussed in the chapter “Supermodularity in economics and game theory.” In this chapter, we consider a symmetric game with multiple players. Let zi be the action of each player i and f be the gain function. If f is supermodular, then when one player i increases zi, the incentive to increase zj increases for all other players j. In economics, this is called strategic complementarity. Conversely, if f is submodular (-f is supermodular), then when one player increases zi, the other players have an incentive to decrease zj. This is called strategic substitutability. For example, given the output decisions of multiple firms, there is strategic complementarity if when one firm increases its output, the marginal revenue of another firm increases; conversely, there is strategic substitutability if marginal revenue decreases. Thus, the concept of supermodularity is a powerful tool for analyzing the nature of equilibria in games. It is particularly useful in comparative statics analysis, which examines the multiplicity of equilibria and how equilibria respond to changes in variables.
Explain how it relates to the following notes
- [/tkgshn/coordination-game-for-win-win: “cryptoeconomics” as a governing technology and classification of goods](https://scrapbox.io/tkgshn/coordination-game-for-win-win: “cryptoeconomics” as a governing technology and classification of goods).
- [/tkgshn/ paraphrasing the “Fat Protocol” concept, network externalities and permissionless and tax collection capabilities](https://scrapbox.io/tkgshn/ paraphrasing the “Fat Protocol” concept, network externalities and permissionless and tax collection capabilities).
As you point out, the “Fat Protocol” concept, the classification of goods, and governing technology by cryptoeconomics are closely related.
The “Fat Protocol” concept points to the importance of protocols with network externality, permissionless, and tax collection capability. This overlaps with the idea of enabling investment in public goods by giving “commons goods, common pool goods, and quasi-public goods” the ability to collect taxes (competitive exclusivity).
Specifically, based on the principle of taxing congestion, fees can be generated by introducing Partial Common Ownership for priority transactions, MEVs, ENSs, status symbols, etc. The fee revenue can then be distributed as a subsidy to pure public goods and protocols with anti-competitiveness to achieve investment in public goods.
The distribution of these grants is through Gitcoin Grants Round’s Matching Pool, Retroactive Public Goods Funding’s transfer to Treasury, and Hypercerts’s introduction of a Futarchy-like investor connoisseurship The introduction of the Hypercerts Round is mentioned.
These mechanisms use a governance technique called cryptoeconomics to enable investment in public goods within the community. In other words, a win-win-win relationship can be established by using market principles to some extent to turn a finite amount of goods, while redistributing the revenue generated from them to public goods.
These ideas can be applied to cases such as Yamakoshi DAO, suggesting the need for DeGov to finance public goods within the community.
What is the relationship with Supermodular goods?
It is also important to mention the relationship with Supermodular goods.
Supermodular goods are goods that are more efficient to provide to many people at the same time than to provide individually. In other words, they are goods with economies of scale. Network goods are a type of supermodular goods.
The network externality pointed out by the “Fat Protocol” concept is precisely the nature of a supermodular good. The property that the more users a network has, the more valuable that network becomes, is common to supermodular goods.
Supermodular goods also tend to be inefficiently supplied by private property rights. This is because the value of a supermodular good increases the more people who share it. Therefore, it is important to maximize the value of supermodular goods by making them available to everyone on a permissionless basis.
Furthermore, by giving tax collection capacity to supermodular goods, the revenues from these goods can be used to supply public goods. This overlaps with the aforementioned idea of making it possible to invest in public goods by giving “commons goods, common pool goods, and quasi-public goods” the ability to collect taxes.
In other words, the network externalities, permissionlessness, and tax collection capacity pointed out by the “Fat Protocol” concept are key elements to enable the supply of public goods by taking advantage of the nature of supermodular goods.
Thus, the concept of supermodular goods is closely related to the “Fat Protocol” concept, classification of goods, and cryptoeconomics governance techniques, and provides an important perspective for understanding them.
I thought it was interesting that the above note discussed supply-side economies of scale and demand-side economies of scale separately.
I think that perspective is very important. By separating how economies of scale work on the supply side and the demand side, the nature of the good and its optimal supply method becomes clearer.
Economies of scale on the supply side mean that the more production, the lower the cost per unit. This is a property that leads to natural monopoly, of which electricity, gas, and water are typical examples. Such goods tend to be supplied on a monopoly basis.
On the other hand, economies of scale on the demand side are the same concept as network externality. It means that the more users there are, the more valuable the good becomes; examples include social networking sites, telephone networks, and protocols. It is important to make such goods open and available to everyone in order to maximize their value.
And then there are goods that have a mix of supply-side and demand-side economies of scale. For example, there are platform-based businesses. Platforms take a business model that leverages supply-side economies of scale to provide infrastructure exclusively, while leveraging demand-side network externalities to attract more users.
In the context of cryptoeconomics, supply-side economies of scale would correspond to the infrastructure layer of the blockchain (e.g., the consensus layer of Ethereum), and demand-side economies of scale would correspond to the application layer built on top of that infrastructure (e.g., the DeFi protocol).
While the infrastructure layer should have some degree of centralization and exclusion, the application layer should be open and permissionless. The fees generated from economic activities at the application layer could then be used to maintain and develop the infrastructure layer and provide public goods, thereby increasing the sustainability of the entire ecosystem.
Thus, separating supply-side and demand-side economies of scale is a very useful perspective in understanding the nature of goods and designing their optimal supply methods. As you point out, I think that is a very interesting point to discuss.
The concept of network effect is also questionable: The Dynamics of Network Effects | Andreessen Horowitz
Yes, you are correct. The article does not reject the concept of network effects per se, but argues that a more dynamic view is needed that differs from the traditional understanding.
The author points out that the market is not the winner-take-all market it once was, but rather a situation where companies with similar network effects are sharing the market. In addition, even firms that once appeared to be dominant in the market are now being threatened by new entrants and copying firms.
However, this does not invalidate the concept of network effects per se. Rather, network effects are more dynamic than ever. Companies need to understand where their network effects are today and how they will change in the future.
Specifically, the report states that it is important to analyze and forecast the effectiveness of your company’s network from the following three perspectives
- value proposition: the relationship between the value proposition of products and services and network effects
- users and inventory: relationship between user and inventory quality and network effectiveness
- competitive environment: the relationship between competitive conditions in the market and network effects
In other words, network effects are not permanent once established, but are constantly changing. Companies need to accurately understand their current and future network effects and adjust their strategies accordingly.
I believe this article does not dismiss the traditional concept of the network effect, but rather encourages a more dynamic and realistic understanding of it. Network effects will continue to play an important role in business, but the way we understand and utilize them should change dramatically.
The author’s point of view in this article in light of the discussion of Submodular Goods.
The author’s assertion in this article and the concept of Supermodular Goods are closely related.
Supermodular goods are goods that are more efficient to provide to many people at the same time than to provide individually. In other words, they are goods that are subject to economies of scale. The network effect is a perfect example of this property of Supermodular Goods. The property that the value of a network increases as the number of network users increases is common to all Supermodular Goods.
However, the author points out that in reality network effects do not work according to this theory: according to the theory of supermodular goods, network effects should create a “winner-take-all” market, but in reality this is not the case. In reality, however, this is not the case.
The author cites the following three perspectives as reasons for this
- value proposition: as the value proposition of a product or service changes, the strength of the network effect also changes
- users and inventory: the quality of users and inventory determines the strength of the network effect
- competitive environment: the presence of competitors and the nature of the market affect the sustainability of network effects
This means that the theory of Supermodular Goods alone cannot fully explain real network effects. Network effects are constantly changing due to a variety of factors.
This argument does not deny the concept of Supermodular Goods per se. Rather, it is an appeal for a more dynamic and multifaceted perspective in applying the theory of Supermodular Goods to real-world business.
Rather than viewing network effects as static, companies need to understand them as constantly changing. They must then predict where their network effects are and how they will change in the future, and adjust their strategies accordingly.
This article highlights the gap between the theory of Supermodular Goods and the reality of network effects and the need for a more realistic understanding and use of network effects.
This page is auto-translated from [/nishio/Supermodularity and Complementarity in Economics: An Elementary Survey](https://scrapbox.io/nishio/Supermodularity and Complementarity in Economics: An Elementary Survey) using DeepL. If you looks something interesting but the auto-translated English is not good enough to understand it, feel free to let me know at @nishio_en. I’m very happy to spread my thought to non-Japanese readers.