Toward An Understanding of Self-Organization of Markets Yougui Wang Department of Systems Science, School of Management, Beijing Normal University, Beijing , China

Aims and Incentives Understanding the performance of markets Starting from the currently dominating perspective on markets Look closely into what really happen in the real world To fill the gap between the classical perspective and reality, we need to move step by step Eventually develop a theoretical model of self-organization of markets

Classical Picture of Markets The market under consideration is for a private good. All product and factor markets are perfect competitive. Production exhibits non-increasing returns to scale. There is no externalities associated with the production and consumption of the good. There is no government intervention of any kind.

Market Equilibrium Price Quantity PePe Supply Demand QeQe

Missing Points in Neo-classical Economics ? The interaction between the individual agents, help to link the optimal choices at individual level and the performance of markets. Demonstrating the movement of the markets, trying to understand not only the equilibrium state of the market but how they get there.

How to reach the equilibrium outcome? Time of trading in the market is discrete. All buyers and sellers of a market come to the market and participate in trade at the same time. A coordinator is needed to set a single price since participants of the market are all price-takers. The coordinator has complete information that means he knows everything for calculating the equilibrium price.

The Reality of Market Performance Trading in markets is performed continuously. Buyers and sellers come to the market at their own will and they make exchange in sequence. The market price is not set by an outsider, instead by participant themselves. No one has complete information of the market. The trade price of the same good alters over different transactions.

New Perspective The two sides of a market should be individualized to replace the overall supply and demand curves. The buyers and sellers in a market can be fully characterized by their willingness prices, which are generated by either rational or irrational choices. The individual rationality of traders in a market erects on the facts they seek for surplus in any particular transaction. The evolution of the market is governed by the interaction among the participants.

Statistical Approach to PEA Yougui Wang, H. E. Stanley, Statistical approach to partial equilibrium analysis, Physica A 388 (2009) Introduction of Willingness Price Restatement of Supply and Demand Formulations of Expected and Realized Market Surplus Prove of Economic Efficiency at Equilibrium

The willingness price of a buyer is defined as the maximum amount that he is willing to pay for one unit of the good.  value, willingness to pay The willingness price of a seller is defined as the minimum amount that he is willing to sell one unit of the good.  cost, willingness to sell Willingness Price

 The necessary condition for buyer to purchase one unit of the good  The necessary condition for seller to produce or sell one unit of the good Rules for Bargain

 There are N sellers in a market whose willingness prices are given respectively and that can be transformed as a continuous distribution function  There are M buyers in a market whose willingness prices are given respectively and that can be transformed as a continuous distribution function Willingness Price Distribution

 The expression of supply function  The expression of demand function Supply and Demand Functions

 Individual Buyer Surplus is the difference between his willingness price and the actual price.  Aggregate Buyer Surplus of a Market is the total surplus that buyers obtain from purchasing a good in the market.  Buyer Surplus is usually called consumer surplus in economics. Buyer Surplus

 Individual Seller Surplus is the difference between the actual price and his willingness price.  Aggregate Seller Surplus of a Market is the total surplus that sellers obtain from selling a good in the market.  Seller Surplus is usually called producer surplus in economics. Seller Surplus

 The aggregate surplus of the sellers can be written as follows  The aggregate surplus of buyers can be written as follows Buyer and Seller Surplus Functions

 Total surplus that buyers and sellers expect to obtain through their potential exchange in a market Market Surplus Being Expected

 When a market is at the equilibrium state, The expected surplus reaches its minimum value. Minimum Expected Surplus at Equilibrium

0PePe P Expected Surplus versus Price

 When a market is at equilibrium, the realized quantity of exchange is equal to quantity supplied or quantity demanded  When a market is not at equilibrium, the realized quantity of exchange is determined by the following equation Short Side Principle

 The overall rationing rate for sellers are defined as the actual quantity of sale divided by quantity supplied.  The overall rationing rate for buyers are defined as the actual quantity of purchase divided by quantity demanded. Overall Rationing Rate

 The overall rationing rate for sellers has the following properties  The overall rationing rate for buyers has the following properties. Overall Rationing Rate

0PePe P 1 Rationing Rate for Sellers Versus Price

0PePe P 1 Rationing Rate for Buyers Versus Price

 Realized quantity of the sellers whose willingness price lies in  The aggregate realized quantity of supply can be written as follows Realized Quantity of Supply

 Realized quantity of the buyers whose willingness price lies in  The aggregate realized quantity of demand can be written as follows Realized Quantity of Demand

 The realized demand is equal to the realized supply for any one case.  The quantity that the buyer purchase equates that the sellers sell Identity of Realized Exchange

 Realized surplus that buyers obtain whose willingness price lies in  Realized surplus that sellers obtain whose willingness price lies in Realized Market Surplus

 Total realized surplus that buyers and sellers obtain through their actual exchange in a market Realized Market Surplus

 When a market is at the equilibrium state, The realized surplus reaches its maximum value. Maximum Realized Market Surplus at Equilibrium

0PePe P Realized Surplus versus Price

Market Self-organization The participants of the market have their own willingness prices as characterizations of them. Every participant has his local information. The participants transact for their goals outside the market and also seek more surplus inside the market. The participant learns by transaction or imitation.

Simulation Results Will the market evolve to ?

Initializations N=1000 sellers with uniformly distributed willingness price of: N=1000 buyers with uniformly distributed willingness price of:

Initializations N=1000 sellers with uniformly distributed willingness price of: N=1000 buyers with uniformly distributed willingness price of: P Q 0 100

Random Matching and Trading Rule In each step, N sellers and N buyers match randomly to make transactions. When seller and buyer are matched to be a trade pair, the seller will ask a price and buyer at the same time bid a price, if they will trade one unit of the good.

Random Matching and Trading Rule Surplus gap Perform Transaction No transaction

Price Adjustments T=3000 times of random matches is called a round. During one round each seller or buyer get 3000 times of random match which they may or, may not make transaction. When a round is over, each seller and buyer will adjust his ask or bid price to get more surplus in the next round. The way to adjust the ask or bid price is to plus or less a certain amount of the former ask or bid price. In this model we choose

Price Adjustments The seller or buyer learns form his former adjustment, if the former adjustment makes more surplus in this round he will adjust the ask or bid price in the same way. If the former adjustment makes less surplus in this round he will adjust the ask or bid price in the opposite way.

Simulation Results C,V~[0,100], δ=1,

Simulation Results C,V uniformly distributed of [0,200], δ=1

Simulation Results C, uniformly distributed of [0,100], v uniformly distributed of [100,200], δ=1 P Q0 200

Analysis: There are two effects of adjust ask or bid price, take a seller as an example. If the seller raises his ask price, he may get more surplus in one transaction. If the seller raises his ask price, he may lose more chance to get transaction. There may be a balance of the two effect.

Analysis: local maximum At equilibrium, each seller or buyer is at his local maximum. Take a seller as example whose cost is C=8.7807, his equilibrium price is while his maximum ask price is around 54.

Analysis: path dependent The reason why the sellers or buyers are locked at a local maximum ask or bid price is that the optimal choice is path dependent. Take one seller whose cost is for example. His total surplus of round 1000 as a function of his ask price can be shown in the figure: (which is the same as the figure of last page)

Analysis: route dependent But at round 1, the seller’s total surplus as a function of his ask price can be shown in this figure:

Analysis: route dependent At round 70, the seller’s total surplus as a function of his ask price can be shown in this figure:

Analysis: route dependent At round 70, the seller’s total surplus as a function of his ask price can be shown in this figure:

Analysis: route dependent At round 90, the seller’s total surplus as a function of his ask price can be shown in this figure:

Analysis: route dependent At round 500, the seller’s total surplus as a function of his ask price can be shown in this figure:

Analysis: route dependent At round 1000, the seller’s total surplus as a function of his ask price can be shown in this figure:

Future Works Framework of continuous time such as double auction process Competitive pressure on individual participants Information channel between individuals and markets Impacts of the mode of connection of participants

Many Thanks!