Market Making with Decreasing Utility for Information

07/30/2014
by   Miroslav Dudík, et al.
0

We study information elicitation in cost-function-based combinatorial prediction markets when the market maker's utility for information decreases over time. In the sudden revelation setting, it is known that some piece of information will be revealed to traders, and the market maker wishes to prevent guaranteed profits for trading on the sure information. In the gradual decrease setting, the market maker's utility for (partial) information decreases continuously over time. We design adaptive cost functions for both settings which: (1) preserve the information previously gathered in the market; (2) eliminate (or diminish) rewards to traders for the publicly revealed information; (3) leave the reward structure unaffected for other information; and (4) maintain the market maker's worst-case loss. Our constructions utilize mixed Bregman divergence, which matches our notion of utility for information.

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1 Introduction

Prediction markets have been used to elicit information in a variety of domains, including business [28, 12, 7, 6], politics [4, 29], and entertainment [25]. In a prediction market, traders buy and sell securities

with values that depend on some unknown future outcome. For example, a market might offer securities worth $1 if Norway wins a gold medal in Men’s Moguls in the 2014 Winter Olympics and $0 otherwise. Traders are given an incentive to reveal their beliefs about the outcome by buying and selling securities, e.g., if the current price of the above security is $0.15, traders who believe that the probability of Norway winning is more than 15% are incentivized to buy and those who believe that the probability is less than 15% are incentivized to sell. The equilibrium price reflects the market consensus about the security’s expected payout (which here coincides with the probability of Norway winning the medal).

There has recently been a surge of research on the design of prediction markets operated by a centralized authority called a market maker, an algorithmic agent that offers to buy or sell securities at some current price that depends on the history of trades in the market. Traders in these markets can express their belief whenever it differs from the current price by either buying or selling, regardless of whether other traders are willing to act as a counterparty, because the market maker always acts as a counterparty, thus “providing the liquidity” and subsidizing the information collection. This is useful in situations when the lack of interested traders would negatively impact the efficiency in a traditional exchange. Of particular interest to us are combinatorial prediction markets [18, 19, 8, 9, 10, 17, 26] which offer securities on various related events such as “Norway wins a total of 4 gold medals in the 2014 Winter Olympics” and “Norway wins a gold medal in Men’s Moguls.” In combinatorial markets with large, expressive security spaces, such as an Olympics market with securities covering 88 nations participating in 98 events, the lack of an interested counterparty is a major concern. Only a single trader may be interested in trading the security associated with a specific event, but we would still like the market to incorporate this trader’s information.

Most market makers considered in the literature are implemented using a pricing function called the cost function [11]. While such markets have many favorable properties [1, 2], the current approaches have several drawbacks that limit their applicability in real-world settings. First, existing work implicitly assumes that the outcome is revealed all at once. When concerned about “just-in-time arbitrage,” in which traders closer to the information source make last-minute guaranteed profits by trading on the sure information before the market maker can adjust prices, the market maker can prevent such profits by closing the entire market just before the outcome is revealed. This approach is undesirable when partial information about the outcome is revealed over time, as is often the case in practice, including the Olympics market. For instance, we may learn the results of Men’s Moguls before Ladies’ Figure Skating has taken place. Closing a large combinatorial market whenever a small portion of the outcome is determined seems to be an unreasonably large intervention.

Second, in real markets, the information captured by the market’s consensus prices often becomes less useful as the revelation of the outcome approaches. Consider a market over the event “Unemployment in the U.S. falls below 5.8% by the end of 2015.” Although there may be a particular moment when the unemployment rate is publicly revealed, this information becomes gradually less useful as that moment approaches; the government may be less able to act on the information as the end of the year draws near. In the Olympics market, the outcome of a particular competition is often more certain as the final announcement approaches, e.g., if one team is far ahead by the half-time of a hockey game, market forecasts become less interesting. Existing market makers fail to take this diminishing utility for information into account, with the strength of the market incentives remaining constant over time.

To address these two shortcomings of existing markets, we consider two settings:

  • a sudden revelation setting in which it is known that some piece of information (such as the winner of Men’s Moguls) will be publicly revealed at a particular time, driving the market maker’s utility for this information to zero; crucially, in this setting we assume that the market maker does not have direct access to this information at the time it is revealed, which is realistic in the case of the Olympics where a human might not be available to input winners for all 98 events in real time;

  • a gradual decrease setting in which the market maker has a diminishing utility for a piece of information (such as the unemployment rate for 2015) over time and therefore is increasingly unwilling to pay for this information even while other information remains valuable.

The sudden revelation setting can be viewed as a special case of the gradual decrease setting. In both cases, we model the relevant information as a variable , representing a partly determined outcome such as the identity of the gold medal winner in a single sports event.

We consider cost-function-based market makers in which the cost function switches one or many times, and aim to design switching strategies such that: (1) information previously gathered in the market is not lost at the time of the switch, (2) a trader who knows the value of  but has no additional information is unable to profit after the switch (for the sudden revelation setting) or is able to profit less and less over time (in the gradual decrease setting), and (3) the market maker maintains the same reward structure for any other information that traders may have. To formalize these objectives, we define the notion of the market maker’s utility (Sec. 2) and show how it corresponds to the mixed Bregman divergence [13, 15] (Sec. 2.5).

For the sudden revelation setting (Sec. 3), we introduce a generic cost function switching technique which in many cases removes the rewards for “just-in-time arbitragers” who know only the value of , while allowing traders with other information to profit, satisfying our objectives.

For the gradual decrease setting (Sec. 4), we focus on linearly constrained market makers (LCMMs) [13], proposing a time-sensitive market maker that gradually decreases liquidity by employing the cost function of a different LCMM at each point in time, again meeting our objectives.

Others have considered the design of cost-function-based markets with adaptive liquidity [21, 22, 24, 23, 3]. That line of research has typically focused on the goal of slowing down price movement as more money enters the market. In contrast, we adjust liquidity to reflect the current market maker’s utility which can be viewed as something external to trading in the market. Additionally, we change liquidity only in the “low-utility” parts of the market, whereas previous work considered market-wide liquidity shifts. Brahma et al. [5] designed a Bayesian market maker that adapts to perceived increases in available information. Our market maker does not try to infer high information periods, but assumes that a schedule of public revelations is given a priori. Our market makers have guaranteed bounds on worst-case loss whereas those of Brahma et al. [5] do not.

2 Setting and Desiderata

We begin by reviewing cost-function-based market making before describing our desiderata. Here and throughout the paper we make use of many standard results from convex analysis, summarized in Appendix A. All of the proofs in this paper are relegated to the appendix. 111The full version of this paper on arXiv includes the appendix.

2.1 Cost-Function-Based Market Making

Let denote the outcome space, a finite set of mutually exclusive and exhaustive states of the world. We are interested in the design of cost-function-based market makers operating over a set of securities on specified by a payoff function , where

denotes the vector of security payoffs if the outcome

occurs. Traders may purchase bundles of securities from the market maker, with denoting the quantity of security that the trader would like to purchase; negative values of are permitted and represent short selling. A trader who purchases a bundle of securities pays a specified cost for this bundle up front and receives a (possibly negative) payoff of if the outcome occurs.

Following Chen and Pennock [11] and Abernethy et al. [1, 2], we assume that the market maker initially prices securities using a convex potential function , called the cost function. The current state of the market is summarized by a vector , where denotes the total number of shares of security that have been bought or sold so far. If the market state is and a trader purchases the bundle , he must pay the market maker . The new market state is then . The instantaneous price of security is whenever well-defined; this is the price per share of an infinitesimally small quantity of security , and is frequently interpreted as the traders’ collective belief about the expected payoff of this security. Any expected payoff must lie in the convex hull of the set , called price space, denoted .

While our cost function might not be differentiable at all states , it is always subdifferentiable thanks to convexity, i.e., its subdifferential is non-empty for each and, if it is a singleton, it coincides with the gradient. Let be called the price map. The set is always convex and can be viewed as a multi-dimensional version of the “bid-ask spread”. In a state , a trader can make an expected profit if and only if he believes that . If is differentiable at , we slightly abuse notation and also use .

We assume that the cost function satisfies two standard properties: no arbitrage and bounded loss. The former means that as long as all outcomes are possible, there are no market transactions with a guaranteed profit for a trader. The latter means that the worst-case loss of the market maker is a priori bounded by a constant. Together, they imply that the cost function can be written in the form , where is the convex conjugate of , with . See Abernethy et al. [1, 2] for an analysis of the properties of such markets.

Example 1.

Logarithmic market-scoring rule (LMSR). The LMSR of Hanson [18, 19] is a cost function for a complete market

where traders can express any probability distribution over

. Here, for any , and where is a 0/1 indicator, i.e., the security pays out $1 if the outcome occurs and $0 otherwise. The price space is the simplex of probability distributions in dimensions. The cost function is , which is differentiable and generates prices . Here is the negative entropy function, .

Example 2.

Square. The square market consists of two independent securities () each paying out either $0 or $1. This can be encoded as with for . The price space is the unit square . Consider the cost function , which is differentiable and generates prices for . Using this cost function is equivalent to running two independent binary markets, each with an LMSR cost function. We have .

Example 3.

Piecewise linear cost. Here we describe a non-differentiable cost function for a single binary security (). Let and , so . The cost function is . It gives rise to the price map such that if , and if , but at , we have , i.e., because of non-differentiability we have a bid-ask spread at . Here, where is a indicator, equal to if true and if false. This market is uninteresting on its own, but will be useful to us in Sec. 3.3.

Input: initial cost function , initial state , switch time ,
Input: update functions ,
Until time :
sell bundles priced using
for the total cost where
let
At time :
After time :
sell bundles priced using
for the total cost where
let
Observe
Pay to traders
PROTOCOL 1 Sudden Revelation Market Makers
Input: time-sensitive cost function ,
Input: initial state , initial time ,
Input: update function
For (where is an unknown number of trades):
at time : receive a request for a bundle
sell the bundle
for the cost
Observe
Pay to traders
PROTOCOL 2 Gradual Decrease Market Makers

2.2 Observations and Adaptive Costs

We study two settings. In the sudden revelation setting, it is known to both the market maker and the traders that at a particular point in time (the observation time) some information about the outcome (an observation) will be publicly revealed to the traders, but not to the market maker. More precisely, let any function on be called a random variable and its value called the realization

of this random variable. Given a random variable

, we assume that its realization is revealed to the traders at the observation time. For a random variable and a possible realization , we define the conditional outcome space by . After observing (where, using standard random variable shorthand, we write for ), the traders can conclude that . Note that the sets form a partition of .

We design sudden revelation market makers (Protocol 1) that replace the cost function with a new cost function , and the current market state (i.e., the current value of in the definition above) with a new market state in order to reflect the decrease in the utility for information about . Such a switch would typically occur just before the observation time. Note that we allow the new cost function as well as the new state to be chosen adaptively according to the last state of the original cost function .

In the gradual decrease setting, the utility for information about a future observation is decreasing continuously over time. We use a gradual decrease market maker (Protocol 2) with a time-sensitive cost function which sells a bundle for the cost at time , when the market is in a state . We place no assumptions on other than that for each , the function should be an arbitrage-free bounded-loss cost function. The market maker may modify the state between the trades.

Protocol 2 alternates between trades and cost-function switches akin to those in Protocol 1. In each iteration , the cost function is replaced by the cost function while simultaneously replacing the state by the state . Crucially, unlike Protocol 1, the cost-function switch here is state independent, so any state-dependent adaptation happens through the state update. 222This simplifying restriction matches our solution concept in Sec. 4, but it could be dropped for greater generality.

At a high level, within each of the protocols, our goal is to design switch strategies that satisfy the following criteria:

  • Any information that has already been gathered from traders about the relative likelihood of the outcomes in the conditional outcome spaces is preserved.

  • A trader who has information about the observation but has no additional information about the relative likelihood of outcomes in the conditional outcome spaces is unable to profit from this information (for sudden revelation), or the profits of such a trader are decreasing over time (for gradual decrease).

  • The market maker continues to reward traders for new information about the relative likelihood of outcomes in the conditional outcome spaces as it did before, with prices reflecting the market maker’s utility for information within these sets of outcomes.

To reason about these goals, it is necessary to define what we mean by the information that has been gathered in the market and the market maker’s utility.

2.3 Market Maker’s Utility

By choosing a cost function, the market maker creates an incentive structure for the traders. Ideally, this incentive structure should be aligned with the market maker’s subjective utility for information. That is, the amount the market maker is willing to pay out to traders should reflect the market maker’s utility for the information that the traders have provided. In this section, we study how the traders are rewarded for various kinds of information, and use the magnitude of their profits to define the market maker’s implicit “utility for information” formally.

We start by defining the market maker’s utility for a belief, where a belief is a vector of expected security payoffs for some distribution over .

Definition 1.

The market maker’s utility for a belief relative to the state is the maximum expected payoff achievable by a trader with belief when the current market state is :

Any subset is referred to as an event. Observations correspond to events . Suppose that a trader has observed an event, i.e., a trader knows that , but is otherwise uninformed. The market maker’s utility for that event can then be naturally defined as follows.

Definition 2.

The utility for a (non-null) event relative to the market state is the largest guaranteed payoff that a trader who knows (and has only this information) can achieve when the current market state is :

Finally, consider the setting in which a trader has observed an event , and also holds a belief consistent with . Specifically, let denote the convex hull of , which is the set of beliefs consistent with the event , and assume . Then we can define the “excess utility for the belief ” as the excess utility provided by over just the knowledge of .

Definition 3.

Given an event and a belief , the excess utility of over , relative to the state is:

Note that in these definitions a trader can always choose not to trade (), so the utility for a belief and an event is non-negative. Also it is not too difficult to see that for any , so the excess utility for a belief is also non-negative.

In Sec. 2.5, we show that given a state and a non-null event , there always exists a (possibly non-unique) belief such that . Thus, a trader with such a “worst-case” belief is able to achieve in expectation no reward beyond what any trader that just observed would receive. We show that these worst-case beliefs correspond to certain kinds of “projections” of the current price onto

. For LMSR, the projections are with respect to KL divergence and correspond to the usual conditional probability distributions. Moreover, for sufficiently smooth cost functions (including LMSR) they correspond to market prices that result when a trader is optimizing his guaranteed profit from the information

as in Definition 2 (see Appendix E). Because of this motivation, such beliefs are referred to as “conditional price vectors.”

Definition 4.

A vector is called a conditional price vector, conditioned on , relative to the state if . The set of such conditional price vectors is denoted

See Appendix F for additional motivation for our definitions of utility and conditioning. With these notions defined, we can now state our desiderata.

2.4 Desiderata

Recall that we aim to design mechanisms which replace a cost function at a state , with a new cost function at a state . Let denote the utility for information with respect to and with respect to , and let and be the respective price maps. In our mechanisms, we attempt to satisfy (a subset of) the conditions on information structures as listed in Table 1.

Price Preserve prices:
.
CondPrice Preserve conditional prices:
.
DecUtil Decrease profits for uninformed traders:
with sharp inequality if .
ZeroUtil No profits for uninformed traders:
.
ExUtil Preserve excess utility:
for all and .
Table 1: Information Desiderata

Conditions Price and CondPrice capture the requirement to preserve the information gathered in the market. The current price is the ultimate information content of the market at a state before the observation time, but it is not necessarily the right notion of information content after the observation time. When we do not know the realization , we may wish to set up the market so that any trader who has observed and would like to maximize the guaranteed profit would move the market to the same conditional price vector as in the previous market. This is captured by CondPrice.

DecUtil models a scenario in which the utility for information about decreases over time, and ZeroUtil represents the extreme case in which utility decreases to zero. These conditions are in friction with ExUtil, which aims to maintain the utility structure over the conditional outcome spaces. A key challenge is to satisfy ExUtil and ZeroUtil (or DecUtil) simultaneously.

Apart from the information desiderata of Table 1, we would like to maintain an important feature of cost-function-based market makers: their ability to bound the worst-case loss to the market maker. Specifically, we would like to show that there is some finite bound (possibly depending on the initial state) such that no matter what trades are executed and which outcome occurs, the market maker will lose no more than the amount of the bound. It turns out that the solution concepts introduced in this paper maintain the same loss bound as guaranteed for using just the market’s original cost function , but since the focus of the paper is on the information structures, worst-case loss analysis is relegated to Appendix H.

In Sec. 3, we study in detail the sudden revelation setting with the goal of instantiating Protocol 1 in a way that achieves ZeroUtil while satisfying CondPrice and ExUtil. Our key result is a characterization and a geometric sufficient condition for when this is possible.

In Sec. 4, we examine instantiations of Protocol 2 for the gradual decrease setting. Our construction focuses on linearly-constrained market makers (LCMM) [13], which naturally decompose into submarkets. We show how to achieve Price, CondPrice, DecUtil and ExUtil in LCMMs. We also show that it is possible to simultaneously decrease the utility for information in each submarket according to its own schedule, while maintaining Price.

Before we develop these mechanisms, we introduce the machinery of Bregman divergences, which helps us analyze notions of utility for information.

2.5 Bregman Divergence and Utility

To analyze the market maker’s utility for information, we show how it corresponds to a specific notion of distance built into the cost function, the mixed (or generalized) Bregman divergence [13, 15]. Let be the conjugate of 333The conjugate is also, less commonly, called the “dual”. The mixed Bregman divergence between a belief and a state is defined as . The conjugacy of and implies that with equality iff , i.e., if the price vector “matches” the state (see Appendix A). The geometric interpretation of mixed Bregman divergence is as a gap between a tangent and the graph of the function (see Fig. 1).

tangent with slope

Figure 1: The mixed Bregman divergence derived from the conjugate pair and measures the distance between the tangent with slope and the value of evaluated at . By conjugacy, the tangent is described by . Note that the divergence is well defined even when is not differentiable, because each slope vector determines a unique tangent.

To see how the divergence relates to traders’ beliefs, consider a trader who believes that and moves the market from state to state . The expected payoff to this trader is . This payoff increases as decreases. Thus, subject to the trader’s budget constraints, the trader is incentivized to move to the state which is as “close” to his/her belief as possible in the sense of a smaller value , with the largest expected payoff when . This argument shows that is an implicit measure of distance used by traders.

The next theorem shows that the Bregman divergence also matches the concepts defined in Sec. 2.3. Specifically, we show that (1) the utility for a belief coincides with the Bregman divergence, (2) the utility for an event is the smallest divergence between the current market state and , and (3) the conditional price vector is the (Bregman) projection of the current market state on , i.e., it is a belief in that is “closest to” the current market state.

Theorem 1.

Let , and . Then

(1)
(2)
(3)

We finish this section by characterizing when ExUtil is satisfied and showing that it implies CondPrice. Recall that and let .

Proposition 1.

ExUtil holds if and only if for all , there exists some such that for all , . Moreover, ExUtil implies CondPrice.

3 Sudden Revelation

In this section, we consider the design of sudden revelation market makers (Protocol 1). In this setting, partial information in the form of the realization of is revealed to market participants (but not to the market maker) at a predetermined time, as might be the case if the medal winners of an Olympic event are announced but no human is available to input this information into the automated market maker on behalf of the market organizer. The random variable and the observation time are assumed to be known, and the market maker wishes to “close” the submarket with respect to just before the observation time, without knowing the realization , while leaving the rest of the market unchanged.

Stated in terms of our formalism, we wish to find functions and from Protocol 1 such that the desiderata CondPrice, ExUtil, and ZeroUtil from Table 1 are satisfied. This implies that traders who know only that are not rewarded after the observation time, but traders with new information about the outcome space conditioned on are rewarded exactly as before. As a result, trading immediately resumes in a “conditional market” on for the correct realization , without the market maker needing to know and without any other human intervention. We refer to the goal of simultaneously achieving CondPrice, ExUtil, and ZeroUtil as achieving implicit submarket closing.

For convenience, throughout this section we write to denote the conditional price space, and to denote prices possible after the observation.

3.1 Simplifying the Objective

We first show that achieving implicit submarket closing can be reduced to finding a function satisfying a simple set of constraints, and defining to return the conjugate of . As a first step, we observe that it is without loss of generality to let be an identity map, i.e., to assume that ; when this is not the case, we can obtain an equivalent market by setting and shifting so that the Bregman divergence is unchanged.

Lemma 1.

Any desideratum of Table 1 holds for and if and only if it holds for .

To simplify exposition, we assume that throughout the rest of the section as we search for conditions on that achieve implicit submarket closing. Under this assumption, Proposition 1 can be used to characterize our goal in terms of . Specifically, we show that ExUtil and CondPrice hold if differs from by a (possibly different) constant on each conditional price space .

Lemma 2.

When , ExUtil and CondPrice hold together if and only if there exist constants for such that for all and .

This suggests parameterizing our search for by vectors . For , define a function

If the sets overlap, is not well defined for all . Whenever we write , we assume that is such that is well defined. To satisfy Lemma 2 with a specific , it suffices to find a convex function “consistent with” in the following sense.

Definition 5.

We say that a function is consistent with if for all .

We next simplify our objective further by proving that whenever implicit submarket closing is achievable, it suffices to consider functions that set to be the conjugate of the largest convex function consistent with for some . To establish this, we examine properties of the convex roof of , the largest convex function that lower-bounds (but is not necessarily consistent with) .

Definition 6.

Given a function , the convex roof of , denoted , is the largest convex function lower-bounding , defined by

where is the set of convex functions , and the condition holds pointwise.

The convex roof is analogous to a convex hull, and the epigraph of is the convex hull of the epigraph of . See Hiriart-Urruty and Lemaréchal [30, §B.2.5] for details.

Example 4.

Recall the square market of Example 2. Let , so traders observe the payoff of the first security at observation time. Then for . For simplicity, let . We have for and for all other . Examining the convex hull of the epigraph of gives us that for all , we have .

As this example illustrates, the roof of is the “flattest” convex function lower-bounding . Given the geometric interpretation of Bregman divergence (Fig. 1), a “flatter” yields a smaller utility for information. This flatness plays a key role in achieving ZeroUtil. Assume that is consistent with , so CondPrice and ExUtil hold by Lemma 2. Following the intuition in Fig. 1, to achieve ZeroUtil, i.e., across all and , it must be the case that for all and , the function values lie on the tangent of with slope . That is, the graph of needs to be flat across the points . This suggests that the roof might be a good candidate for . This intuition is formalized in the following lemma, which states that instead of considering arbitrary convex consistent with , we can consider which take the form of a convex roof.

Lemma 3.

If any convex function is consistent with then so is the convex roof . Furthermore, if satisfies ZeroUtil or DecUtil then so does .

3.2 Implicit Submarket Closing

We now have the tools to answer the central question of this section: When can we achieve implicit submarket closing? Lemma 1 implies that we can assume that is the identity function, and Lemmas 2 and 3 imply that it suffices to consider functions that set to the conjugate of for some . What remains is to find the vector that guarantees ZeroUtil. As mentioned above, ZeroUtil is satisfied if and only if lies on the tangent of with the slope for all and . This implies that for all and and some constant . The specific choice of does not matter since is unchanged by vertical shifts of the graph of . For convenience, we set , which makes the tangents of and with the slope coincide. This and Lemma 2 then yield the choice of , with

(4)

for all and any choice of . The resulting construction of can be described using geometric intuition. First, consider the tangent of with slope equal to the current market state . For each , take the subgraph of over the set and let it “fall” vertically until it touches this tangent at the point . The set of fallen graphs for all together describes and the convex hull of the fallen epigraphs yields .

Defining using this construction guarantees ZeroUtil, but CondPrice and ExUtil are achieved only when is consistent with . Conversely, whenever the three properties are achievable, this construction produces a function consistent with . This yields a full characterization of when implicit submarket closing is achievable.

Theorem 2.

Let be defined as in Eq. (4). CondPrice, ExUtil, and ZeroUtil can be satisfied using Protocol 1 if and only if is consistent with . In this case, they can be achieved with as the identity and outputting the conjugate of .

3.3 Constructing the Cost Function

Theorem 2 describes how to achieve implicit submarket closing by defining the cost function output by implicitly via its conjugate . In this section, we provide an explicit construction of the resulting cost function, and illustrate the construction through examples.

Fixing , for each define a function . Each function can be viewed as a bounded-loss and arbitrage-free cost function for outcomes in . The conjugate of each coincides with on (and is infinite outside ). The explicit expression for is described in the following proposition.

Proposition 2.

For a given with conjugate , define as in Eq. (4) and let . The conjugate of can be written . Furthermore, for each , .

At any market state with a unique , the price according to lies in the set . When is not unique, the market has a bid-ask spread. The addition of ensures that the bid-ask spread at the market state contains conditional prices across all . To illustrate this construction, we return to the example of a square.

Example 5.

Consider again the square market from Examples 2 and 4 with . One can verify that for . Prop. 2 gives


In switching from to we have effectively changed the first term of our cost from a basic LMSR cost for a single binary security to the piecewise linear cost of Example 3, introducing a bid-ask spread for security 1 when ; states have . The market for security 1 has thus implicitly closed; as the new market begins with , any trader can switch the price of security 1 to 0 or 1 by simply purchasing an infinitesimal quantity of security 1 in the appropriate direction, at essentially no cost and with no ability to profit.

The example above illustrates our cost function construction, but does not show that is consistent with as required by Theorem 2. In fact, it is consistent. This follows from the sufficient condition proved in Appendix G.2. Briefly, the condition is that does not contain any price vectors that can be expressed as nontrivial convex combinations of vectors from multiple .

In Appendix G.3, we show that this sufficient condition applies to many settings of interest such as arbitrary partitions of simplex and submarket observations in binary-payoff LCMMs (defined in Sec. 4), which were used to run a combinatorial market for the 2012 U.S. Elections [14].

A case in which the sufficient condition is violated is the square market with , where and but . This particular example also fails to satisfy Theorem 2 (see Appendix G.1), but in general the sufficient condition is not necessary (see Appendix G.4).

4 Gradual Decrease

We now consider gradual decrease market makers (Protocol 2) for the gradual decrease setting in which the utility of information about a future observation is decreasing continuously over time. We focus on linearly constrained market makers (LCMMs) [13], which naturally decompose into submarkets. Our proposed gradual decrease market maker employs a different LCMM at each time step, and satisfies various desiderata of Sec. 2.4 between steps.

As a warm-up for the concepts introduced in this section, we show how the “liquidity parameter” can be used to implement a decreasing utility for information.

Example 6.

Homogeneous decrease in utility for information. We begin with a differentiable cost function in a state . Let , and define , and . is parameterized by the “liquidity parameter” . The transformation guarantees the preservation of prices, i.e., . We can derive that , and , so, for all , . In words, the utility for all beliefs with respect to the current state is decreased according to the multiplier .

This idea will be the basis of our construction. We next define the components of our setup and prove the desiderata.

4.1 Linearly Constrained Markets

Recall that is the payoff function. Let be a system of non-empty disjoint subsets forming a partition of coordinates of , so . We use the notation for the block of coordinates in , and similarly and . Blocks describe groups of securities that are treated as separate “submarkets,” but there can be logical dependencies among them.

Example 7.

Medal counts. Consider a prediction market for the Olympics. Assume that Norway takes part in Olympic events. In each, Norway can win a gold medal or not. Encode this outcome space as . Define random variables equal to 1 iff Norway wins gold in the th Olympic event. Also define a random variable representing the number of gold medals that Norway wins in total. We create securities, corresponding to 0/1 indicators of the form for and for . That is, for and for . A natural block structure in this market is with submarkets corresponding to the and .

Given the block structure , the construction of a linearly constrained market begins with bounded-loss and arbitrage-free convex cost functions with conjugates and divergences for each . These cost functions are assumed to be easy to compute and give rise to a “direct-sum” cost with the conjugate and divergence .

Since decomposes, it can be calculated quickly. However, the market maker might allow arbitrage due to the lack of consistency among submarkets since arbitrage opportunities arise when prices fall outside  [1]. is always polyhedral, so it can be described as for some matrix and vector . Letting denote the th column of , arbitrage opportunities open up if the price of the bundle falls below . For any , the bundle presents an arbitrage opportunity if priced below .

A linearly constrained market maker (LCMM) is described by the cost function . While the definition of is slightly involved, the conjugate has a natural meaning as a restriction of the direct-sum market to the price space , i.e., . Furthermore, the infimum in the definition of is always attained (see Appendix D.1). Fixing and letting be a minimizer in the definition, we can think of the market maker as automatically charging traders for the bundle , which would present an arbitrage opportunity, and returning to them the guaranteed payout . This benefits traders while maintaining the same worst-case loss guarantee for the market maker as [13].

Example 8.

LCMM for medal counts. Continuing the previous example, for submarkets , we can define LMSR costs . For the submarket for , let and use the LMSR cost . The submarkets for and are linked. One example of a linear constraint is based on the linearity of expectations: for any distribution, we must have . This places an equality constraint on the vector , which can be expressed as two inequality constraints (see Dudík et al. [13, 14] for more on constraint generation).

4.2 Decreasing Liquidity

We now study the gradual decrease scenario in which the utility for information in each submarket decreases over time. In the Olympics example, the market maker may want to continuously decrease the rewards for information about a particular event as the event takes place.

We generalize the strategy from Example 6 to LCMMs and extend them to time-sensitive cost functions by introducing the “information-utility schedule” in the form of a differentiable non-increasing function with . The speed of decrease of controls the speed of decrease of the utility for information in each submarket. (We make this statement more precise in Theorem 3.)

We first define a gradual decrease direct-sum cost function which is used to define a gradual decrease LCMM, and a matching as follows:

When considering the state update from time  to time , the ratio has the role of the liquidity parameter in Example 6. The motivation behind the definition of is to guarantee that , which turns out to ensure that remains the minimizer and the prices are unchanged. The preservation of prices (Price) is achieved by a scaling similar to Example 6, albeit applied to the market state in the direct-sum market underlying the LCMM.

This intuition is formalized in the next theorem, which shows that the above construction preserves prices and decreases the utility for information, as captured by the mixed Bregman divergence, according to the schedules . We use the notation and write for the divergence derived from .

Theorem 3.

Let be a gradual decrease LCMM, let and . The replacement of by and by satisfies Price. Also,

(5)

for all , where and are defined by , and .

The first term on the right-hand side of Eq. (5) is the sum of divergences in submarkets , each weighted by a coefficient which is equal to one at and weakly decreases as grows. The divergences are between and the state resulting from the arbitrager action in the direct-sum market. The second term is non-negative, since , and represents expected arbitrager gains beyond the guaranteed profit from the arbitrage in the direct-sum market. The only terms that depend on time  are the multipliers . Since they are decreasing over time, we immediately obtain that the utility for information, , is also decreasing, with the contributions from individual submarkets decreasing according to their schedules .

When only one of the schedules is decreasing and the other schedules stay constant, we can show that the excess utility and conditional prices are preserved (conditioned on ), and under certain conditions also DecUtil holds.