Computational economies: private and differential information

A working framework in economic theory is the concept of an abstract economy. An abstract economy is defined as a topological space⁹¹, where a set of economic agents face scarcity, and the

91 The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. A topological space, also called an abstract topological space, is a set X together with a collection of open subsets τ that satisfies the four conditions:

(i) The empty set ∅ ∈ τ and the universe X ∈ τ

(ii) If Cα ∈ τ for all α ∈ A, then Uα∈ACα ∈ τ (the union of a finite/arbitrary number of sets in τ is also in τ) (iii) If Cj ∈ τ for all 1 ≤ j ≤ n, then ∩nj=1Cj ∈ τ (the intersection of a finite/arbitrary number of sets in τ is

also in τ)

Then we say that τ is a topology on X and that (X, τ) is a topological space. These axioms are designed so that the traditional definitions of open and closed intervals of the real line continue to be true. The key idea is that a topological space is not a set of points, along with a set of neighborhoods for each point. The relationship between the points comes from how they're laid out: something additional is needed to know how the points are "arranged". A topology is one sort of this additional information.

Note that a topology doesn't tell the difference between the curve <|> and the curve <S>; that

26 notion of scarcity is susceptible to be expressed in symbolic terms⁹². Abstract economies are theoretical constructions usually used to prove theorems about economic equilibrium and stability⁹³. Regarding price behaviour however, they are not immediately interesting since the agents have all the information in the economy at all times, meaning price is just the independent variable for demand and supply functions, which act together to reach market equilibria. In order to make abstract economies informationally interesting, it is necessary to introduce asymmetries in the information the agents possess, acquire and exploit in their benefit. This is achieved by allowing agents to have/acquire private information, either as an initial endowment and/or as a result of market transaction analysis⁹⁴.

Private information is usually defined in terms of game theory. In it most basic form, private information is the knowledge of a player of the convenience of selecting a particular action in response to a particular event in the game. Let’s say there is a finite set I of players. Each player i in I first observes the realization of a random event zi ∈ Z in the game. As a result, chooses his own action from a nonempty measurable subset Di(zi) of a countable complete metric space Ai (A is the set of all combinations of all players’ moves, Ai being the corresponding subset of all possible moves for player i). The payoff of each player i is given by the utility function ui: A × Xi → R (X being the set of all possible payoffs to all possible moves). Let µ be a probability measure on Ω, representing the fact that for each player and each pair (event, payoff) there is a utility function assigning a finite payoff. Therefore, a finite game with private information is a family Γ = (I, ((Zi, Xi), (Ai, Di), ui)i ∈ I, µ). To transform the game into an abstract economy, two restrictions are to be considered. First, each agent (player) has a n-vector function (g1, g2,.., gn) representing a pure strategy profile. So her chose of a response to an event in the economy is constrained to those responses fitting her strategy.⁹⁵ For each i ∈ I, let the constraint correspondence αi be the set Ai

adjusted to the agent’s strategy. Moreover, the agent has preferences, which are expressed as a correspondence Pi. Agents should have bounded factual knowledge and computational capability⁹⁶, which require extending the economy to allow such skills. This is why extended abstract economies

information comes from different formalisms (e.g. the concept of a "curve in the Euclidean plane").

Sources:

‘Topological Space’. Available at: http://mathworld.wolfram.com/TopologicalSpace.html

‘Topological space’. Wikipedia. Available at: https://en.wikipedia.org/wiki/Topological_space . Accessed online 20.06.2017.

Körner, T. W. “Metric and Topological Spaces”. August, 2015. Tom Körner's Home Page, available at:

https://www.dpmms.cam.ac.uk/~twk/ .

‘Intuition behind open set in topology’. Available at:

https://math.stackexchange.com/questions/1430405/intuition-behind-open-set-in-topology

92 For a formal definition of an abstract economy, please refer to Fläm, Sjur D. “Abstract economies and games”. Soochow Journal of Mathematics, Vol.5, p. 55. December, 1979.

93 Richter, Michael and Rubinstein, Ariel. “Back to Fundamentals: Equilibrium in Abstract Economies”.

American Economic Review, Vol. 105, No. 8, pp. 2570-2594. August, 2015.

Singh, S.P.; Tarafdar, E. and Watson, B. “A generalized fixed point theorem and equilibrium point of an abstract economy”. Journal of Computational and Applied Mathematics, Volume 113, Issues 1–

2, pp. 65–71. January, 2000.

94 No reference is made here to qualitative aspects of information, which in real world financial markets is an issue due to noise and other factors, such as cyclic behaviour of business activity, among other sources of bias.

95 Equilibrium in abstract economies is possible even in an infinite dimension strategy space, provided that there is more than one agent. See Kim, Taesung. “Equilibria in Abstract Economies with a Measure Space of Agents and with an Infinite Dimensional Strategy Space”. Journal of Approximation Theory 56, pp.256-266. 1989.

96 Simon, Herbert A. The Sciences of the Artificial. Cambridge MIT Press, 1969.

27 are often referred as generalized games⁹⁷. An abstract economy (or a generalized game) with private information and a countable space of actions is defined as Γ = (I, (Zi, (Ai, αi, Pi), ui)i ∈ I, µ) ⁹⁸.

Existence of private information does not mean the agents cannot share the same knowledge over time, if strategies are subject to refinement according to experience, eventually the agents may end up with very similar private information (depending on the preferences). The next step is to allow for asymmetries in information the agents possess, creating differential information economies (DIE). A DIE is the abstraction needed to support algorithmically determined prices. The simpler possible interpretation of a differential information economy would be as follows. The economy extends over two periods, T0 and T1. In the first period there is uncertainty over states of the world in the subsequent period. In T0, agents make contracts (agreements) either before the state of nature is realized (ex ante) or after they have received a signal as to what is the event containing the realized state of nature (interim). In T1, agents carry out the previously made agreements and consumption takes place. So the alternative for the agents in T0 is whether to make their decision with or without a certain amount of information⁹⁹.

A formal definition of a differential information economy is:

Let Ω denote a finite set of states of the world, and let R^C denote the Euclidian commodity space. An exchange economy with differential information is a set E = {(Xi, ui, Fi, ei, µ) : i = 1, 2,..., n}, where:

1. Xi = R^C > 0, the consumption set of agent i,

2. ui : Ω × Xi → R is the (random) utility function of agent i, 3. Fi is a partition of Ω denoting the private information of agent i,

4. ei : Ω → Xi is the (random) initial endowment of agent i, where each ei(·) is Fi –measurable, 5. µ is the common prior (probability distribution) of all agents.

Notice that the allocation of commodities should be measurable with respect to the partition F of each agent. It is both feasible and not susceptible to be improved upon by any redistribution of their initial endowments based on net trades which are, in turn, measurable with respect to agents’ private information. Technically, the private information Fi is a sub‒σ-algebra of F¹⁰⁰.

Now, let’s check that a differential information economy effectively supports price features which are relevant to algorithmic pricing:

a) price formation mechanisms should be able to take in a wide range of items to be priced, and b) allow for price manipulation through coalitions, as it was discussed in Case study 1.1 – Price

manipulation in the Amazon marketplace.

Well, regarding letter ‘a’, the mathematical definition of a DIE does not impose restrictions on the commodity set as long as the agents are able to consume and derive utility from that consumption.

The economic importance of intangible assets like brands, reputation, corporate culture and efficient teamwork as sources of value, has increased as agents’ sentiment play a role in the information

97 For a review of the key assumptions, principles, and applications of Generalized Game Theory, please refer to Burns, Tom R. and Roszkowska, Ewa. ‘Generalized game theory: assumptions, principles, and elaborations grounded in social theory’. Studies in Logic, Grammar and Rhetoric 8 (21), 2005. For a proof of an equilibrium existence theorem for an abstract economy (generalized game), please check Tan, Kok-Keong and Yuan, Xian-Zhi. “Approximation method and equilibria of abstract economies”. Proceedings of the American Mathematical Society, Volume 122, Number 2.

1994.

98 Patriche, Monica. “Existence of equilibrium for an abstract economy with private information and a countable space of actions”. 2013.

99 The interpretation and definition of a differential information economy is taken from:

Allen, Beth and Yannelis, Nicholas C. “Differential information economies: Introduction”. Economic Theory 18, pp. 263–273. 2001.

100 Patriche, Monica. “A Bayesian Abstract Economy with a Measure Space of Agents”. Abstract and Applied Analysis Volume 2009.

28 economy¹⁰¹. Moreover, the role of knowledge management and information systems in developing new market niches, in creating and distributing innovative instruments, and in increasing product

‘virtualization’ suggests that information and communication systems have also became a significant part of the markets’ processes and the agents decision-making procedures¹⁰². Regarding letter ‘b’, DIEs allow cooperative solution arrangements, bargaining and coalitions (situations where groups of agents organize themselves for the purpose of bargaining with the rest of the players to increase their payoffs, i.e. political parties, unions, and cartels) to exist, therefore providing valid alternatives to the standard rational expectations equilibrium.¹⁰³

The relevance of differential information economies to algorithmic pricing has been established.