Prediction markets are an increasingly popular primitive, currently being used to facilitate speculation on the probability of various political, geopolitical and economic events. While this use case is promising, as demonstrated by the increasingly high TVL of protocols such as Polymarket, it only unlocks a fraction of what prediction markets have to offer.
Decision markets are how prediction markets become useful for making decisions. The event probabilities which prediction markets currently elicit can sometimes be useful for informing decisions, however they are not optimised for this purpose.
Prediction markets only reveal the probability of an event, whereas decision markets reveal how different actions would impact the outcome being predicted. For example, a prediction market might reveal that p(GDP increasing year over year) = 0.4 (where p() denotes probability), whereas a decision market could tell us that conditional on electing Ron Paul, p(GDP increasing) = 0.9, whereas only 0.06 if Kamala Harris were to be elected (figures are for illustrative purposes only).
While the output of a prediction market can be useful, the conditional probabilities provided by conditional prediction markets are much more actionable. In this case, they could be used to conclude that voting for Ron Paul is the superior action, if one’s goal is a higher GDP.
All the above however is still framed in the context of politics, which despite being the domain in which prediction markets have excelled, is unlikely to be where decision markets create the most value. Politics-focused prediction markets are popular because people enjoy betting on matters which they have tribal or ideological connections to, given that the target customer base of prediction markets, such as Polymarket, is speculators. The target customer base of decision markets, however, will be information buyers; those willing to pay to discover which action the market believes they should take in order to maximise success according to a given metric.
The goal of decision markets is to elicit from market participants what the expected impact of an action is on a selected metric. While this might seem simple, given the below high level outline, there are several confounding, alignment and manipulation risks which are important to consider and potentially mitigate and which are the focus of this article.
Mechanism overview
Here’s how decision markets work in the context of a DAO evaluating funding proposals (where funding a proposal is considered an action):
For a batch of funding proposals, we create conditional tokens:
- A “Yes” token for each proposal that pays out based on the metric if that proposal is executed
- A single “No” token for the batch that pays out based on the metric if none of the proposals are executed
These tokens use a scalar design where the payout is proportional to the value of a composite metric. The key tokens are:
- \textsf{Long}^{\text{yes}}_i: Redeems for a value proportional to the metric if proposal i is executed
- \textsf{Long}^{\text{no}}: Redeems for a value proportional to the metric if no proposal in the batch is executed
The relative price of \textsf{Long}^{\text{no}} and each \textsf{Long}^{\text{yes}}_i reveal what the market expects each action’s impact to be on the selected metric (subject to caveats, analysis of which will constitute the remainder of this report). From here onwards score(action) will be used to refer to \frac{\textsf{Long}^{\text{yes}}_i}{\textsf{Long}^{\text{no}}} for a given action, where a higher score(action) indicates an action is assessed as being more favourable in terms of the metric. The value of score(action) can be used to select which actions to take, on the basis of the ratio between each action’s cost and its score, i.e. its cost:benefit ratio.
Implementation of the above requires selection of an AMM design or other mechanism for facilitating the exchange of shares and construction of the composite metric on which to evaluate actions. A decision rule also must be selected, which is an algorithm for deciding which actions to take, given the predicted impact on the metric and magnitude of each action’s respective cost.
For some further reading on decision market design, see:
- Conditional Funding Markets
- amoveo-docs/basics/msrs_and_prediction_markets.md at 3747c9c7b330a03531da0cbeb111fafac6839f81 · zack-bitcoin/amoveo-docs · GitHub
- Prediction market design for betting on many highly improbable events - Economics - Ethereum Research
- Practical Futarchy Setup - Economics - Ethereum Research
- https://timroughgarden.github.io/fob21/reports/ZLRL.pdf
- Possible Futarchy Setups - Economics - Ethereum Research
- https://mason.gmu.edu/~rhanson/combobet.pdf
- Manifesto/Manifesto.pdf at main · metaDAOproject/Manifesto · GitHub
Confounding
Decision markets can suffer from confounding because they are trying to use market-implied conditional probabilities as a proxy for measuring the impact of an action on the value of a metric. Decisions are made based on the relative prices of \textsf{Long}^{\text{yes}} and \textsf{Long}^{\text{no}} shares, in each action’s market. Any factors which speculators consider when pricing these shares, aside from their expectations of an action’s impact, are confounding variables. Confounding variables distort the price with information which is irrelevant to maximising the value of the metric, hence making the decision market prices less useful.
Confounding occurs when the variable you are measuring the impact on (the dependent variable) is affected by variables other than the variable you are actually trying to measure the impact of (the independent variable). You might want to measure the effect of A on B but end up accidentally measuring something different due to one or multiple of the below three types of confounding:
Common cause confounding
- due to the effect that C has on both A and B
- where C is what is known as a “common cause” variable, due to it influencing both A and B, making it seem as though they are causally connected
An illustration of common cause confounding is that ice cream sales are correlated with shark attacks, but one should be careful not to conclude from this that recently having ingested ice cream causes sharks to pay them special attention:
Higher consumption of ice cream
↑
Warmer weather <---------- confounding common cause variable
↓
More people swimming at the beach
↓
More shark attacks
Mediation confounding
- when you end up measuring the the effect of A on D and the effect of D on B
- where D is a “mediating” variable, acting as a “bridge” between A and B
Mediation applies when a variable exists which partially or fully mediates the causal relationship between the dependent and independent variables. Mediation can confound measurements when the mediating variable is not accounted for, causing its impacts to be incorrectly attributed to the independent variable directly impacting the dependent variable. An example of this is if one concluded that propeller rotation is directly responsible for aircraft propulsion, without considering the movement of air (a mediating variable), and therefore decided to use propellers as a propulsion mechanism for spacecraft.
Rotation of propeller
↓
Aerodynamics resulting from presence of atmosphere <---------- confounding mediating variable
↓
Propulsion of aircraft
Reverse causality confounding
- where you measure the effect of B on A, instead of measuring the effect of A on B
- i.e. the direction of causation is opposite to what you thought it was
- so what you think is the independent variable is actually the dependent variable and what you think is the dependent variable is actually the independent variable
An example in the context of reverse causality would be concluding that painkillers cause injuries due to consumption of painkillers and injuries being highly correlated:
Injury
↓ <---------- confounding reverse causal pathway
Consumption of painkillers
Strategy: Standard futarchy setup
The most straightforward setup is to base action decisions directly on the decision market’s output and decision rule. This approach is typically what people mean when they refer to “decision markets.” It has two major advantages:
- Decisions are informed by the information generated by the prediction markets.
- Risk of confounding variables is mitigated
At a high level, this setup achieves these benefits by forcing all potential confounds through a single, narrow bottleneck: score(action). This represents the decision market’s expectation of the impact of taking vs not taking an action, on the value of the metric.
The above mentioned confounding mechanisms are only possible if the choice of action is influenced by a confounding variable. As a result, the relevance of score(action) to confounding is hard to overstate, as any confounding variable must go through score(action) in order to influence the choice of action, given that actions are chosen according to their respective score(action) values. score(action) is the sole thing standing between the standard futarchy setup and one where confounding is impossible, due to e.g. actions being selected at random.
Below are some causal diagrams which depict different types of confounding in the standard futarchy setup.
Ideal situation (no confounding):
expected impact of action
↓
score(action)
↓
p(action is taken)
Common cause:
p(action is taken)
↑
score(action)
↑
hypothetical "common cause" confounding variable
↓
expected impact of action
Reverse causality:
p(action is taken)
↓ <-------- hypothetical confounding reverse-causal pathway
score(action)
↓
expected impact of action
Mediation:
expected impact of action
↓ ↓
↓ hypothetical "mediating" variable
↓ ↓
score(action)
↓ ↓
p(action is taken)
Mediation confounding
Mediation confounding can occur in this setup if a “mediating variable” exists, creating an additional causal path between expected impact of action and p(action is taken). Mediating variables become problematic when they’re not consistently present, causing the measured correlation to diverge from the real-world correlation where the mediating variable may be absent. For instance, in the propeller example, the mediating variable (atmosphere) becomes an issue when the propulsion technology is applied in the vacuum of space.
In this futarchy setup, a mediating variable exists that is present during measurement but not during execution, potentially distorting score(action). This variable is the decision rule, which selects actions based on their score(action) values. While intended to choose the most promising action, it inadvertently increases the correlation between p(action is taken) and expected impact of action by mediating an additional causal path between them (illustrated in the diagram below). Consequently, it biases score(action) upwards for all actions, as score(action) effectively measures the correlation between p(action is taken) and expected impact of action.
expected impact of action
↓ ↓
↓ decision rule
↓ ↓
score(action)
↓ ↓
p(action is taken)
This tendency of the standard futarchy setup to overestimate the benefit of an action would be of minimal concern if it applied to all actions equally, as it would then at least preserve the ranking of actions according to their expected benefit. Unfortunately however, this distortion affects some markets more than others, and hence can meaningfully impact the relative attractiveness of actions, according to their decision market scores.
Impact of new information
The reason for this asymmetry is that the magnitude of the distortion expected to result from a mediating variable is proportional to its expected variance over the period of the market. If the mediating variable is not expected to change, it is consequently also not expected to increase the correlation between expected impact of action and p(action is taken), and hence will not impact score(action). As a result, the more variance expected in the value of score(action) (and hence in the output of the decision rule), for a given action, the more the decision market will optimistically bias its estimate of the action’s benefit.
greater variance in score(action)
↓
greater variance of output of decision rule
↓
greater correlation between expected impact of action and p(action is taken)
↓
higher value of score(action)
↓
action is (artificially) more likely to be selected
In other words, the more new information expected to be revealed about an action, the more positively biased its decision market score will be. This can also be thought of as the result of shares conditional on an action being taken having a convex payoff, like options, resulting in positive gamma, and hence positive vega (volatility increases probability of reaching higher points on the convex payoff function), which signifies a positive sensitivity to changes in volatility.
Another framing of this is that if you think an action’s market-implied score is too high, you will be less inclined to correct the mispricing, by buying \textsf{Short}^{\text{yes}}, if you think the market will converge to your belief prior to the market closing. This is because if the market-implied score converges (downwards) to your belief, the action is less likely to be taken, in which case your position is worth $0 due to being conditional on the action being taken. So the more you expect the market to update towards your belief, hence the higher the volatility, the less attractive your short position will seem and the more attractive a long position will seem.
Manipulation risk
This is challenging, as it makes it difficult to compare market prices for different actions, due to them potentially being distorted to different degrees by this optimism bias. An economic vulnerability also potentially arises, if malicious proposal creators exploit this to increase the decision market score of their proposal, by attempting to create an expectation that new information pertinent to it will soon be revealed.
For example, consider a proposal for developing an improved risk modeling suite for the DAO’s treasury management. The proposal creator might strategically mention that they are working on a new approach to risk assessment, the details of which will be finalized during the initial phase of the project. They might hint at preliminary tests showing promising results in predicting market volatility, without providing specific data. This creates an expectation of important new information to be revealed, potentially inflating their decision market score due to the optimism bias.
proposer hints at important announcement
↓
greater uncertainty
↓
greater variance of output of decision rule
↓
greater correlation between expected impact of action and p(action is taken)
↓
higher value of score(action)
↓
action is (artificially) more likely to be selected
Mitigation strategies may involve any of the following:
- Measuring the expected (or perhaps even the actual) variance of each market, then arithmetically adjusting each market’s price to control for this variable, using techniques from option pricing.
- Equalising/limiting variance across markets by penalising or rejecting proposals from creators who claim that important information is yet to be revealed by the time their proposal’s decision market has begun.
- Automatically extending markets if their volatility soon before resolution is above a certain threshold, so that the decision market price used for action selection is free from the optimistic bias present prior to the revelation of new information.
- Ensuring traders are aware of the economic vulnerability, so they are attuned to and can discount markets where they perceive the creator to be trying to create artificial uncertainty around the impact of the proposal.
- Limiting the duration of markets such that it is implausible a meaningful amount of new information will be revealed over the period of the market.
Reverse causality
Reverse causality is also only partially mitigated by this default decision market configuration. Specifically, reverse causality remains possible via the same channel through which common cause confounding is possible: score(action). If there is a causal path from expected impact of action to to action is selected, confounding due to reverse causality can occur. Analysis into what may cause reverse causality and or common cause confounding in decision markets is left for future research.
Strategy: Decision randomisation
An alternative strategy, capable of completely eliminating confounding, is to simply select which actions to take at random. The reason this works is akin to the reason patients are randomly selected to be in either the experimental or control arms of a study. It ensures no confounding differences between participants in the two arms of the study remain, so the only difference is that participants of the experimental arm received the intervention while those in control did not.
expected impact of action
↓ ↓
↓ random decision rule <------- action choice now randomised, hence "deactivating" this confounding pathway
↓ ↓
score(action)
↓ ↓
p(action is taken)
In the diagram above, randomising the decision rule causes it to no longer have a disparate impact on actions according to how high their score(action) is. As a result, it effectively removes the confounding pathway.
The problem with this approach is that it is very expensive, because it requires actions to be taken at random regardless of their expected outcome. It is ironic that for decision market prices to not be confounded, they must not be used to make decisions, as the actions must be taken at random. Fortunately it is possible to avoid almost all confounding, while still being able to use decision market prices to inform decisions.
Strategy: Randomise, sometimes
The “randomise, sometimes” strategy addresses confounding in decision markets by occasionally selecting actions at random. It works by cancelling the market 90% of the time, making all shares worth $0. In the remaining 10% of cases, an action is randomly selected and the markets are resolved based on the results of these random actions. Trader rewards are then multiplied by 10 to compensate for the 90% cancellation rate.
This approach eliminates confounding because traders price shares solely based on the random selection scenario, while the non-random 90% doesn’t affect pricing due to market cancellation. Importantly, this strategy allows decision-makers to use market predictions to inform choices 90% of the time, while still maintaining unbiased market prices. It effectively separates the price formation process from actual decision-making, preserving the market’s informational value while preventing feedback loops that could distort prices.
expected impact of action
↓
score(action)
↓
Random selection (10% chance)
/ \
/ \
↓ ↓
Random action Use score(action)
↓ ↓
Resolve market Cancel market
↓
10x rewards
↓
Informs future pricing
(No confounding pathway)
This strategy has two notable advantages:
- 90% (or some other high proportion) of the time, you can take the action which the decision markets predict will most positively impact the metric. We cancel the market in these cases to prevent the non-random action selection process from contaminating the price.
- No confounding is possible, because traders price the decision market shares only according to the scenario where the action is selected at random, because in all other cases, the market is cancelled.
It however also has the following challenges:
- 10% of the time, we will need to select an action at random which means this strategy is limited to cases where taking the worst possible action wouldn’t pose an existential risk to the entity using decision markets to make decisions. In some cases, it may be possible to eliminate a large amount of downside risk from this strategy by vetting actions for obvious issues before they are traded on, however false positives and false negatives are inevitable, and the trust assumptions required to implement vetting may be prohibitive for some use cases.
- The variance of returns is higher which, according to the Kelly criterion, means each trader’s optimal allocation to decision market shares will be lower. This can also be seen as manifestation of volatility drag. As a result, information is less efficiently elicited from informed market participants, due to their optimal position size being lower, causing them to less quickly correct mispricings.
Given the above three discussed decision market strategies, a continuous trade-off space emerges between 1) the need to sometimes choose actions at random 2) having to accept a small amount of confounding and 3) market accuracy (due to higher variance).
Accuracy
/ \
/ \
/ \
Confounding-------Randomisation
Note:
At the beginning of this article a decision market was referenced with the goal of determining the impact of various presidential candidates on GDP. In practice, though, this market is problematic and more so a conditional prediction market than a decision market, as it doesn’t attempt to isolate causality by avoiding confounding. The democratic process used to elect presidents makes no attempt to avoid common cause confounding or reverse causality, as it simply was not designed with decision markets in mind.
