Algorithmic Game Theory and Internet Computing Vijay V. Vazirani Georgia Tech Extending General Equilibrium Theory to the Digital Economy

Irving Fisher, 1891 Defined a fundamental market model Special case of Walras’ model

Several buyers with different utility functions and moneys.

W.l.o.g. assume 1 unit of each good

Several buyers with different utility functions and moneys. Equilibrium prices

Digital Goods Can be stored in the memory of a computer.

Pricing of Digital Goods From first principles! Music, movies, video games, … cell phone apps., …, web search results, …

Pricing of Digital Goods Music, movies, video games, … cell phone apps., …, web search results, … Once produced, supply is infinite!!

An interesting observation Any Pareto optimal allocation must give a copy of each digital good to each agent!

An interesting observation Any Pareto optimal allocation must give a copy of each digital good to each agent! Hence usual notion of equilibrium is not applicable!

Jain & V., 2010: Redefine “supply” and “demand” for digital goods and “map” digital economy onto AD-model. Proof of existence of equilibrium via Kakutani fixed point theorem. Efficient algorithm for one case.

Pricing of Digital Goods Music, movies, video games, … cell phone apps., …, web search results, …, even ideas! Once produced, supply is infinite!!

Idiosyncrasies of Digital Realm Staggering number of different goods, belonging to same genre, available with equal ease. E.g., iTunes has 11 million songs! App Store has 300,000 iPhone apps!

Game-Theoretic Assumptions Full rationality, infinite computing power

Game-Theoretic Assumptions Full rationality, infinite computing power e.g., song A for $1.23, song B for $1.56, …

Game-Theoretic Assumptions Full rationality, infinite computing power, and infinite patience! e.g., song A for $1.23, song B for $1.56, …

Game-Theoretic Assumptions Full rationality, infinite computing power: not meaningful! e.g., song A for $1.23, song B for $1.56, … Cannot price songs individually!

Uniform pricing has naturally emerged! Forerunners of digital goods, e.g., CD’s, DVD’s etc. iTunes  Started with $0.99 for each song.  Now has 3 categories: $0.69, $0.99 and $1.29

Model: categories of digital goods E.g., jazz music, children’s movies, classic movies. Uniform pricing of all goods in a category Generic good in a category: “song”

Categories of digital goods How to price categories? How to allocate goods from categories? Notion of equilibrium?

More idiosyncrasies … A person desiring 2 songs wants 2 different songs! 2 copies of same song no better than 1 copy!

More idiosyncrasies … A person desiring 2 songs wants 2 different songs! 2 copies of same song no better than 1 copy! Making a loaf of bread vs. making a copy of a song – very different!

Yet more … People have widely different liking for songs in a category & different people may have totally different likings!

Model: total orders Assume g categories of digital goods. : Set of songs in category j. These songs “define” category j. For each digital category j, each agent i has a total order over all songs in

Market Model Assume 1 conventional good: bread (1 unit), and g digital categories ( songs in category j). Each agent i has a coarse utility function over g digital categories and bread, and has money

Optimal bundle for i, given prices p Coarse allocation: Using and find optimal amounts: Detailed allocation: For each digital category, j, i gets her most favorite songs from, as given by

Copyright Law Only original owner/producer of a song can sell copies.

A basic difference In AD-model, any is a valid bundle. In our model: Suppose  {A, E} not a valid bundle  {A, B} is valid.

A basic difference In AD-model, any is a valid bundle. In our model: Suppose  {A, E} not a valid bundle  {A, B} is valid. Hence, our model cannot be “reduced” to AD-model.

The “mapping”

Each molecule of bread can be bought by at most one buyer. is replaced by Each song can be bought at most once by each buyer.

Supply and demand: conventional goods No deficiency: No surplus:

Supply and demand: digital goods No deficiency: No surplus:

Equilibrium (p, x) s.t. Each agent, i, gets optimal bundle (detailed allocation).  Does not demand more than songs in category j Market clears, i.e., all bread sold & at least full 1 copy of each song sold.

Equilibrium (p, x) s.t. Each agent, i, gets optimal bundle (detailed allocation).  Does not demand more than songs in category j Market clears, i.e., all bread sold & at least full 1 copy of each song sold. Cannot replace 1 by any other integer!

Market Model

Market Model Show: Equilibrium exists!

Theorem (Jain & V., 2010): Equilibrium exists. “market maker” consumes surplus songs, and produces songs to cover deficiency. At fixed point, MM does nothing!

Extensions of existence theorem Arrow-Debreu (exchange) model Introduce production in both models

Open More complex, realistic models, e.g., complementarity in production. Economics of ads in digital marketplaces. Model creation of new categories, extinction of old categories.

Algorithmic questions Efficient algorithms for remaining cases, especially first case with arbitrary no. of conventional goods + digital categories. Experimental verification & applications, e.g., pricing of new digital goods.

and more … Pricing of financial advice – need to sell to a few people at high price. Pricing of “innovative ideas” Pricing of drugs Other natural models that don’t need to rely on game theoretic assumptions of full rationality & infinite computing power.

Arrow-Debreu model with production Agent can use 8 hrs/day to make bread or write a song in any category (or any combination) total order over “defining” songs. Are owned by agents.

Arrow-Debreu model with production Feasible production of each agent is a convex, compact set in Agent i’s earning:  no. of units of bread produced  no. of copies of songs (owned or produced by i) sold Agent spends earnings on optimal bundle.

Equilibrium (p, x, y) s.t. Each agent, i, gets optimal bundle & “best” songs available in each category. Each agent, k, maximizes earnings, given p, x, y (-k)  Note: this aspect resembles Nash equilibrium! Market clears.

Market clears All bread produced is sold For each category j,  At least 1 full copy of each song in sold.  No agent demands more songs than those produced and those in

A question Q: What if a talented song writer wants to sell her songs at a premium? A: Negligible cost of copying. Hence important to capture market share! Our model automatically rewards best song writers.

Important clarification Our model is not postulating an entity like iTunes that prices categories. The market decides customary price for each category via supply & demand. Anyone ignoring market pricing norms is taking a risk of being ignored.