Decision-makers often adaptively influence downstream competitive agents' behavior to minimize their cost, yet in doing so face critical challenges: $(i)$ decision-makers might not *a priori* know the agents' objectives; $(ii)$ agents might *learn* their responses, introducing stochasticity and non-stationarity into the decision-making process; and $(iii)$ there may be additional non-strategic environmental stochasticity. Characterizing convergence of this complex system is contingent on how the decision-maker controls for the tradeoff between the induced drift and additional noise from the learning agent behavior and environmental stochasticity. To understand how the learning agents' behavior is influenced by the decision-maker’s actions, we first consider a decision-maker that deploys an arbitrary sequence of actions which induces a sequence of games and corresponding equilibria. We characterize how the drift and noise in the agents' stochastic algorithms decouples from their optimization error. Leveraging this decoupling and accompanying finite-time efficiency estimates, we design decision-maker algorithms that control the induced drift relative to the agent noise. This enables efficient finite-time tracking of game theoretic equilibrium concepts that adhere to the incentives of the players' collective learning processes.

We study auction design in a setting where agents can communicate over a censorship-resistant broadcast channel like the ones we can implement over a public blockchain. We seek to design credible, strategyproof auctions in a model that differs from the traditional mechanism design framework because communication is not centralized via the auctioneer. We prove this allows us to design a larger class of credible auctions where the auctioneer has no incentive to be strategic. Intuitively, a decentralized communication model weakens the auctioneer's adversarial capabilities because they can only inject messages into the communication channel but not delete, delay, or modify the messages from legitimate buyers. Our main result is a separation in the following sense: we give the first instance of an auction that is credible only if communication is decentralized. Moreover, we construct the first two-round auction that is credible, strategyproof, and optimal when bidder valuations are $α$-strongly regular, for $α> 0$. Our result relies on mild assumptions -- namely, the existence of a broadcast channel and cryptographic commitments.

Constant function market makers (CFMMs) are the most popular type of decentralized trading venue for cryptocurrency tokens. In this paper, we give a very general geometric framework (or 'axioms') which encompass and generalize many of the known results for CFMMs in the literature, without requiring strong conditions such as differentiability or homogeneity. One particular consequence of this framework is that every CFMM has a (unique) canonical trading function that is concave, homogeneous, and nondecreasing, showing that many results known only for homogeneous trading functions are actually fully general. We also show that CFMMs satisfy a number of intuitive and geometric composition rules, and give a new proof, via conic duality, of the equivalence of the portfolio value function and the trading function. Many results are extended to the general setting where the CFMM is not assumed to be path-independent, but only one trade is allowed. Finally, we show that all 'path-independent' CFMMs have a simple geometric description that does not depend on any notion of a 'trading history'. Many of the results here depend on a type of duality for concave, homogeneous, and nondecreasing functions which may be of independent interest.