the intersection of MEU and risk based trader models. Therefore it becomes inter- esting to verify whether the Abernethy et al. [2014] result can be extended to general risk measure based traders, which would give a convergence guarantee to the risk- measure based trader model in multi-period markets (i.e., the model considered by Hu and Storkey [2014]).
Chen et al. [2007], Dimitrov and Sami [2008] and Chen et al. [2009] also give con- vergence and non-convergence properties of MSR market prices to “Perfect Bayesian Equilibrium” (which is different from the Walrasian equilibrium considered in this thesis). They consider strategic play by traders and situations where traders receive different information.
2.5
Repeated markets
Finally, I survey what is known about the dynamics of repeated equilibrium mar- kets and the interpretation of Bayesian wealth updates in equilibrium markets. In Chapter 5, I will present preliminary work that relates wealth updates in sequen- tial markets (with mini-trading) to wealth updates in equilibrium markets and open questions about repeated sequential markets.
Consider a setup of repeated predictions (like a series of weather predictions) where equilibrium markets are run for each prediction instance and the traders are paid off when the outcome of the current prediction is known, before moving on to the next prediction instance. This would mean that traders will have an updated wealth before moving on to the next prediction. The following results show that equilibrium prediction markets perform Bayesian wealth updates for the traders.
Beygelzimer et al. [2012] show that in binary outcome prediction markets with Kelly betters (log utility based MEU traders), the equilibrium market redistributes traders’ wealth according to Bayesian law when the outcome of the market is known. Note that a Log utility trader make demands W.(pn/π∗n) spending his entire wealthW,
given beliefs p and equilibrium prices π∗ (adopting the notation for multiple out-
come markets as in Section 2.1.4.1). Since he spends his entire wealth, if outcome
y = i ∈ [N] occurs, his wealth would be updated to W.(pi/π∗i). Plugging in the
equilibrium prices π∗n = ∑Kk=1Wk.pkn (which is the wealth-weighted mean beliefs),
we get that for trader j ∈ [K], wealth would be updated to Wj.p
j i
∑K k=1Wk.pki
if outcome
y = i occurs. This corresponds with a Bayesian wealth update for the trader, so that the posterior probability of choosing the trader j in Bayesian model averag-
§2.5 Repeated markets 32 ing is P(j|y = i) = P(y=i|j).P(j) P(y=i) = Wj.pj i ∑K k=1Wk.pki
(if the prior probability of trader j is
P(j) =Wj/(∑K
k=1Wk)and the likelihood of outcome given trader isP(y=i|j) = pki).
Storkey et al. [2012] generalize this result to the class of iso-elastic utility based MEU traders, using the “effective belief” of traders as the likelihood of outcome given a particular trader. Considering the form of demands in the case of iso-elastic utility based MEU traders, the “effective belief” of a trader is considered as a belief adjust- ment by a trader considering his own beliefs and the market prices. The demands of an iso-elastic trader is given bydn(π∗) = W(pn/π
∗
n)β
∑N
n=1π∗n.(pn/π∗n)β
(when the trader spends his entire wealth for trading), which can be written asdn(π∗) =W.(En(π∗)/πn∗), where En(π∗) = (pn/π
∗
n)β.πn∗
∑N
n=1π∗n.(pn/π∗n)β ∈ ∆N is considered as the “effective belief” of a trader
[Storkey et al., 2012].
Further, the equilibrium price for iso-elastic traders can also be written as a wealth- weighted mean of “effective beliefs” of traders [Storkey et al., 2012], i.e., the equi- librium price πn∗ = {∑Kk=1Wk
Zk.(pkn)β}(1/β), where Zk = ∑Nn=1qkn. pkn/qkn β
can be re- written as πn∗ = ∑Kk=1Wk.Enk(π∗). So the wealth updates and market prices in case
of iso-elastic traders correspond with Bayesian wealth updates and Bayesian model averaging when the “effective beliefs” are used as the traders’ beliefs.
Chapter3
A Common Interpretation of
Equilibrium and Sequential
Markets
This chapter presents the work on interpreting the instantaneous price of sequential markets as stochastically minimizing (via Stochastic Mirror Descent (SMD)) the same objective as its Walrasian equilibrium, thus giving a common understanding of in- stantaneous and equilibrium prices via an optimisation point of view [Premachandra and Reid, 2013]. This would naturally extend the existing machine learning interpre- tation of equilibrium markets (i.e., as producing the optimum aggregation minimis- ing divergence-based distances of traders with different beliefs) to sequential mar- kets. Also understanding sequential markets as SMD would enable the use of same to achieve the optimum aggregation (which happens to be the equilibrium price) us- ing a market approach instead of having to impose the use of convex optimisation techniques to solve for equilibrium. This would enable the use of prediction markets to extend machine learning models to more complex scenarios (see Storkey [2011]), since markets can be used for solving generalised aggregation problems where each trader is related to different types of “distances”.
Theorem 1 in Section 3.3 relates equilibrium prices to those obtained from sequential mechanisms acting upon the same stochastic market. The mini-trading mechanism introduced in Section 3.4 can be seen as an implementation of SMD with properties (e.g., stability, bounded loss) that make it desirable for finding equilibrium prices. The closest work to the contribution in this chapter is by Frongillo et al. [2012] who first interpreted sequential markets as performing a SMD and established a connec- tion between sequential markets and equilibrium analysis using limiting conditions on sequential markets. The interpretation of sequential markets as SMD as given in