Section 3.3 is a variation of their result for a more broader class of trades and a more restrictive class of markets. The second part of the Theorem shows that the solution of the stochastic optimisation problem is the equilibrium price of the market, thus establishing a connection between sequential markets and equilibrium without using limiting conditions.
Before presenting the contribution I will briefly introduce the two types of markets we consider (Section 3.1) and present concepts related to convex analysis (Section 3.2) which makes it possible to interpret the stochastic price update of sequential markets via stochastic mirror descent.
3.1
Stochastic Markets
We are interested in mechanisms for aggregating trader beliefs about a single future event with N possible outcomes (e.g., who will win an election or horse race) and will label them 1, . . . ,N. We will assume that the outcomes are mutually exclusive and complete (i.e., exactly one of 1, . . . ,Ncan occur). We consider prediction markets where there areNtypes of contracts to be traded – one for each of theNoutcomes – that pay $1 if outcomenoccurs and nothing otherwise. These markets are known as “complete markets”.
The relationship between equilibrium and sequential markets we obtain is under the assumption of stochastically drawn demands. The concept of drawing traders from a distribution to analyse markets is not new. In equilibrium analysis, it analy- ses the expected behaviour of the market as a whole. In sequential market analysis, traders are usually drawn i.i.d. to determine the order or trading. For e.g., Manski [2004] and Wolfers and Zitzewitz [2006]’s equilibrium market analysis and Frongillo et al. [2012][Theorem 1]’s sequential market analysis consider traders (who make de- mands dependant on their beliefs and wealth) drawn from a joint distribution over beliefs and wealths.
Storkey [2011] considers traders who make demands based on their utility function, so two traders with the same beliefs and wealth may not act similarly even when given the same market prices. We adopt this type of market, in the sense that a trader’s demand (as defined in Section 2.1.4) is also dependant on the trader model, thus allowing for markets with “inhomogeneous” traders as in Storkey [2011] and Storkey et al. [2012]. So we model markets as a distribution of demands (instead of as a distribution of beliefs and wealths).
§3.1 Stochastic Markets 35
There are two broad classes of market mechanisms considered: 1. Equilibrium markets (introduced in Section 2.2)
2. Sequential cost function based markets (introduced in Section 2.3.3)
Since demand operators (introduced in Section 2.1.4) are a general way of describing trader behaviour, we model a market for both scenarios as a setM ⊂ D of demand operators. We further assume that the demand operators for a market are drawn i.i.d. from some distribution σ over D and refer to the distribution σ as a stochastic market.
A stochastic analysis of markets provide much greater power than a fixed point model associated with an equilibrium analysis. Focusing on the expected quantities of a stochastic setting helps in relating to fixed point equilibria. However when it comes to uniqueness and convergence, the stochastic setting is much more powerful (i.e., conditions for the uniqueness of distributional equilibria for stochastic systems are much cleaner and easier to establish than for fixed point analyses).
Next I will present the formal definition of market equilibrium and formalise the price update mechanism in sequential markets under stochastically drawn demands. 3.1.1 Equilibrium Market Mechanisms
Classically, the(Walrasian) equilibrium pricefor a market is a fixed price at which there is no excess demand for any good [Varian, 2009; Vazirani, 2007]. Similar to Storkey [2011]; Storkey et al. [2012], we consider a market with more than two types of con- tracts and assume that traders come to the market with only a cash amount (and no previously purchased contracts). We define the equilibrium pricefor a stochastic market σ to be the price πσ∗ ∈ ∆N such that the expected demand of the market in
response to thefixed-pricepricing function Ππσ∗ is equal to the total spendings of the
traders. Formally, the equilibrium price satisfies Ed∼σ h dΠπσ∗ i =Ed∼σhΠπ∗σ dΠπσ∗ i .1 (3.1)
Or using the shorthand notation for fixed-price demands Ed∼σ
h
dΠπ∗σ i
§3.1 Stochastic Markets 36
This is similar to the definition used by Storkey et al. [2012] where the expected de- mand for any outcome equals the total wealth of the traders, assuming without loss of generality that traders spend their entire wealth for trading. If instead traders spend $0, then we get Frongillo et al. [2012]’s definition that the expected demand equals zero. This also corresponds with thebudget conservationprinciple used by Lay and Barbu [2010, 2011].
Also if the expected demand for all outcomes equal anyD∈Rsuch thatEd∼σ[d(πσ∗)] = D.1, then we have that the total spendings also equalD(Equation 3.3), thus satisfying the equilibrium condition.
Ed∼σ[hπσ∗,d(π∗σ)i] =hπσ∗,Ed∼σ[d(πσ∗)]i= D (3.3)
Equilibrium prices for a non-stochastic market withKtraders with demandsd1, . . . ,dK
can be obtained via a distribution that puts mass 1/K over those K demands. The question of the existence of equilibrium prices is complex but resolved [Arrow and Debreu, 1954].
Normally the equilibrium is defined for MEU traders who maximise a concave func- tionU(x) :RN →R, so that the equilibrium condition returns an equilibrium price that also satisfies all the traders (since traders calculate their demands to maximise their expected utilities). Arrow and Debreu [1954] gives an existance proof for equi- librium for MEU traders under convexity assumptions (which holds when the in- dividual utilities are concave). Uniqueness of equilibrium is only satisfied under gross substitutability (or other additional conditions). This is indeed satisifed for a market of individual securities, but may not be satisfied more generally1. Also Lay and Barbu [2010, 2011] gives mild conditions on the demands functions (namely that the demand for outcomenis continuous and strictly decreasing in πn) for a unique
equilibrium. In this chapter and rest of the thesis, we don’t necessarily assume MEU traders and a unique equilibrium.
In our Theorem 1, we assume that traders make demands to optimise some value (not necessarily the expected utility), and use the assumption that demands are potential- based (see Section 3.3.2). This yields a form equivalent to the above definition of market equilibrium (Equation 3.2), thus being able to define the equilibrium market as minimising an objective that is related to the potential-based representation of
1See our Lemma 2 which shows that MEU traders return a unique demand for a given fixed price