We focus here on Nash equilibrium NE , in which each player selects a strategy maximizing its expected payoff, given the strategies of the other players. In this study, we model the market as a symmetric game , which means that each player has the same available strategy set, and each has the same payoff as a function of its own strategy and the set of other-agent strategy choices. In other words, payoffs depend only on the number of agents playing each strategy, not on the identities of the agents playing them.
We also focus attention on symmetric NE , in which each player selects the same possibly mixed strategy in equilibrium. Note that even for a symmetric profile, on any given run the players will generally choose different strategies a consequence of independent selection from the mixture , and will also generally have heterogeneous private valuations. To analyze a game, we apply EGTA in an iterative manner, interleaving exploration of the profile space with analysis of the empirical game model induced by average payoffs in simulation. We start by simulating all the symmetric pure-strategy profiles, where a single strategy is shared by all players.
Exploration then spreads through their neighbors, that is, those profiles related by single-agent deviations. For this reason, the game is incomplete at any point during the EGTA process, as some profiles have been empirically evaluated whereas others have not.
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As payoffs from simulation are incorporated into the empirical game, we analyze each successive intermediate game model by computing mixed equilibria for each complete subgame. A normal-form subgame is the game obtained by restricting the set of strategies, and a complete subgame is defined as a subgame for which all profiles have been evaluated by simulation.
The symmetric Nash equilibria of the complete subgames are candidates for equilibria in the full game. If we can identify a strategy in the full strategy set that beneficially deviates from the candidate, we say the candidate is refuted. A candidate profile is confirmed as an NE when all possible deviations have been evaluated, and none are beneficial. We confirm or refute each candidate by evaluating deviations to strategies outside their subgames.
If a candidate is refuted, we construct a new subgame by adding the best response to its support, and proceed to explore the corresponding subgame. We simulate additional profiles for a game until we have confirmed at least one symmetric NE, evaluated every pure-strategy symmetric profile, and pursued with some degree of diligence every equilibrium candidate encountered.
More specifically, we continue to refine the empirical game with additional simulations until the following conditions are met: 1 at least one equilibrium is confirmed,. The procedure described above seeks to either confirm or refute the equilibrium candidates detected in our exploration of the strategy space.
As we are not able to exhaustively search the entire profile space, however, additional qualitatively distinct equilibria are always possible. In addition, the equilibria we find are subject to refutation by other strategies outside the specified set. Our search process described above attempts to evaluate all promising equilibrium candidates e.
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Even with a moderate number of players, the game size number of possible strategy profiles grows exponentially with the number of players and strategies, rendering analysis of the full game computationally infeasible. As such, we apply aggregation to approximate the many-player games as games with fewer players: We employ the technique of deviation-preserving reduction DPR developed by Wiedenbeck and Wellman to construct a reduced-game approximation of the full game. DPR preserves the payoffs from single-player, unilateral deviations, and maintains in the reduced game the same proportion of opponents playing each strategy as in the full game.
In a deviation-preserving reduced game, each player views itself as controlling one full-game agent and views the other-agent profile in the reduced game as an aggregation of all other players in the full game. Although the equilibrium approximations obtained via DPR are not guaranteed estimates, DPR has been shown to produce good approximations in other games Wiedenbeck and Wellman, DPR defines reduced-game payoffs in terms of payoffs in the full game as follows.
To isolate the ramifications of market fragmentation, we consider two consolidated market configurations in addition to the two-market model: a CDA and a frequent call market. A frequent call market eliminates latency arbitrage opportunities, as the periodic clearing mechanism makes it impossible to gain or exploit informational advantages over other market participants within the clearing interval.
In exploring the relationship between trader behavior and market structure, we are interested in the following performance characteristics:. Allocative efficiency. Total surplus welfare is our key measure of market performance.
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Welfare indicates how well the market allocates trades according to underlying private valuations. Markets are liquid to the extent they maintain availability of opportunities to trade at prevailing prices. We measure execution time by the interval between order submission and transaction for orders that eventually trade. Execution time is potentially important to investors for many reasons, including the risk of changes in valuation while an order is pending, the effect of transaction delay on other contingent decisions, and general time preference.
To facilitate direct comparison, in each run we set the clearing interval of the call market to equal the NBBO update latency. Each row of the table describes the market configurations included as indicated by the plus symbol in evaluating a given market feature.
The four market configurations are the two-market model 2M both with and without LA, the consolidated CDA, and the frequent call market. We evaluate and compare the performance of the four market structure configurations two-market model with and without LA, CDA, and frequent call market within three distinct environments. For the fragmented cases, an equal proportion of background traders is assigned primary affiliation with each market in a model. In defining our environments, we selected environment parameters that generate sufficient arbitrage opportunities and also replicate the original findings for fixed-strategy, non-equilibrium comparisons from our previous study Wah and Wellman, To do so, we explored a number of environments, varying the number of traders, trading horizon length, degree of mean reversion, and variance in both the fundamental and private values.
We selected the environments reproducing the qualitative effects previously observed as the starting point for the extended strategic analysis of the current study.
All bids have single-unit quantities, and we assume zero transaction costs. ZI strategy combinations included in empiricalgame-theoretic analysis.
Definition of 'Arbitrage'
The games in a given environment include one single-market CDA game which is independent of latency , and one game for each of the other three market configurations two fragmented cases and one with periodic clears per latency setting simulated. At latency 0, the frequent call market is equivalent to the CDA, and the two models with fragmentation are equivalent as there are no arbitrage opportunities at zero latency. The empirical games for each environment include a consolidated CDA, which is independent of latency, and one game at each latency setting for the other three market configurations.
At latency 0, the fragmented models are equivalent to the CDA, as there are no arbitrage opportunities. Latency here with regards to the frequent call market indicates the length of the clearing interval. For each equilibrium, we estimated background-trader surplus, as well as LA profit if applicable, by sampling profiles according to the equilibrium mixture, and running simulations per sampled profile 50, full-game simulations in total.
Values presented are the average over strategies in the profile, weighted by mixture probabilities. Surplus values are means from thousands of simulations of the full game, where strategies are randomly sampled from the equilibrium mixed-strategy profile. The numeric column headings give R max values for the ZI strategies. The total surplus of the two-market model without LA, as well as that of the single CDA market an unfragmented continuous-time market , generally exceeds that of the two-market model with LA, whether or not the profits of LA are counted.
This holds across the three environments. In other words, the latency arbitrageur takes surplus away from the background investors, and the amount it deducts exceeds the gross trading profit it accrues. Total surplus in the two-market 2M model, both with and without a latency arbitrageur, and in the consolidated CDA market, for the three environments. Each point reflects the average over 50, simulation runs of the maximum-welfare equilibrium for each market configuration and latency setting.
The intuition behind this result lies in differences in the orders selected to trade. A continuous market will match two orders to trade immediately, regardless of whether this transaction improves allocative efficiency. The LA, by matching orders across the two fragmented markets, is prone to facilitate some inefficient trades that would not execute in the two-market model without latency arbitrage.
Welfare differences that arise from changes in order sequencing in continuous markets. The order book initially has two sell orders.
Two buy orders arrive, with different sequencing, over the course of two time steps. In the top scenario, the buy order at price 9 arrives before the buy order at 15, resulting in total surplus of 5 from two trades assuming traders submit orders priced at their valuations. In the bottom scenario, the buy order at price 15 arrives first, which results in a more efficient transaction with a higher surplus of 7 than the alternate scenario. Each pair of green circles indicate orders that have matched and traded at a given moment in time. As a result, background traders in these two environments shade more in response to the LA.
This can be seen by higher R mid values in the NE found. Prior work by Zhan and Friedman has shown that strategicallydetermined levels of bid shading can mitigate inefficient trades in CDAs. The infinitely fast arbitrageur immediately exploits arbitrage opportunities that arise from incorrectly routed orders; these LA trades tend to be inefficient, which contributes to the lower overall welfare observed in the two-market model with LA.
Increased bid shading in the low mean reversion environments can alleviate some of these inefficiencies, which improves background-trader surplus and reduces LA profits.
Note that when latency is zero, the two fragmented models and the CDA market in Fig. The NBBO is always correct if there is no delay, so it is not possible for any latency arbitrage opportunities to emerge. It follows that the various market configurations at zero latency produce similar total surplus in equilibrium. As for the case without latency arbitrage, it may seem counterintuitive that welfare in the two-market model without LA is higher than in the consolidated CDA in some environments.
It turns out that fragmentation can actually provide a benefit for continuous markets. The separated markets are less likely to admit inefficient trades i. LA can defeat this benefit by ensuring that any orders that would match in the central CDA also trade in the fragmented case, albeit with LA rather than with a counterpart investor. This is largely due to the different strategies selected in equilibrium in this environment; traders tend to shade their bids less i. The improvement in execution time is at best approximately 1—2 time steps, however, which is generally insignificant to non-HFtraders.
Execution time in the two-market 2M model, both with and without a latency arbitrageur, and in the consolidated CDA market, for the three environments. Execution time is the difference between bid submission and transaction times. Each point reflects the average over 50, simulation runs of the maximum-welfare equilibrium for the market configuration and latency setting. Traders in the other two environments, however, do not shade more in equilibrium in the two-market model without LA. In these cases, the fastest execution is achieved in the consolidated CDA, which makes sense given the absence of both communication latencies and thinness induced by fragmentation.