# Artifical Markets based on SDG

The Specker Derivative Game (SDG) is generalized to a much larger family of artificial derivative markets parameterized by a maximization problem with an objective function with range in [0.1]. According to Mark Mueller from the Algorithmic Trading Group at GMO (http://www.gmo.com/America/About/People/_Departments/AlgorithmicTrading.htm) our artificial markets are innovative and quite different from other artificial markets including the artificial markets studied at the Santa Fe Institute. The hope is that some of the SDG-based artificial markets are interesting models of real markets and can be used to improve algorithmic traders in real markets.

So far (2008) artificial markets have not have a strong impact on algorithmic trader design but this might change in the future.

We provide a brief description of the SDG-based artificial markets. Choose a maximization problem Max and define a family of predicates to define subsets of the instances of Max. The predicates are defined by a language (which is defined by a context-free grammar and a checker that selects the legal predicates).

## Derivatives as Predicates

A derivative in SDG(Max) consists of a triple: a predicate in the allowed family of predicates, a price in [0,1] and a player. The players offer and buy derivatives. Buying a derivative gives you the following two rights: (1) to receive from the owner raw material R (an instance of maximization problem Max) satisfying the predicate and (2) upon finishing the raw material R at quality q (trying to find the maximum solution), you receive q in [0,1] from the owner of the derivative.

## Maximum Satisfiability Example

Example: Choose as maximization problem: Maximum Satisfiability. Instances = weighted sets of clauses. Objective: satisfy the maximum fraction of weighted clauses. The predicates are pairs of pairs ((l1,p1), (l2,p2)), where each inner pair defines an allowed clause type. (l1,p1) defines clauses of length l1 with p1 positive literals. (((2,0),(1,1)), 0.55, Ahmed) defines a derivative created by Ahmed where we allow only the following two kinds of clauses: Clauses of length 2 containing two negative literals and clauses of length 1 containing one positive literal. In this case the break-even price for (((2,0),(1,1)) is 0.618 = GoldenRatio. The GoldenRatio = ((sqrt(5)-1)/2) is a phase transition point separating P from NP.

It will be interesting to study the artificial markets for different maximization problems and different notions of predicates. In the "traditional" instantiations (using MAXSAT and MAXCSP), the anlysis of the game relies crucially on a closure property: The set of instances defined by a predicate must be closed under symmetrization. This property supports efficient precise computation of the break-even points for each derivative.

## Relevance of Artificial Markets?

A fundamental problem is how to price a derivative. If the trading robots play optimally, each derivative has a break-even price. But the trading robots are resource constrained and might not be able to find the raw material that is best for them or they might not be able to find a good approximate solution to the maximization problem and both facts might drive away the break-even price from the optimal value, creating more opportunities to make money. It is a form of exploiting the weaknesses of the other trading robots if you know those weaknesses well enough. This happens also to real-life derivatives where their value might not depend only on the fundamental values of the involved assets but also on public opinion.

Artificial Markets (I don't know the quality/relevance of those links):

```http://lfe.mit.edu/research/artificial-markets.htm
http://www.ccs.neu.edu/home/lieber/evergreen/specker/artificial-markets.html
http://people.brandeis.edu/~blebaron/wps/sfisum.pdf
http://econpapers.repec.org/paper/col000094/004616.htm
```