Abstract
Discrete event games are discrete time dynamical systems whose state transitions are discrete events caused by actions taken by agents within the game. The agents’ objectives and associated decision rules need not be known to the game designer in order to impose structure on a game’s reachable states. Mechanism design for discrete event games is accomplished by declaring desirable invariant properties and restricting the state transition functions to conserve these properties at every point in time for all admissible actions and for all agents, using techniques familiar from state-feedback control theory. Building upon these connections to control theory, a framework is developed to equip these games with estimation properties of signals which are private to the agents playing the game. Token bonding curves are presented as discrete event games and numerical experiments are used to investigate their signal processing properties with a focus on input-output response dynamics.
Supported by the Research Institute for Cryptoeconomics at WU Vienna in collaboration with BlockScience, Inc.
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Notes
- 1.
- 2.
We adopt notation and conventions from the signal processing literature throughout, opting for a unified exposition at the potential risk of cross-disciplinary “notation collision”.
- 3.
It may be the case that, for agent a,
$$ U(x_t, a) \equiv U(\hat{x}_{a,t}, a), $$i.e. the agent’s state space is sufficient to define their actions (this would be the case, for example, in a game where every agent has their own action set, or where every agent conditions only upon their own private information). In what follows we allow for full conditioning on \(x_t\).
- 4.
Naturally there may still be an implicit unique mapping from \(u_t\) to a, as is the case with sending Bitcoin, [30].
- 5.
While ostensibly this model assumes a strict ordering of actions, this is a consequence of the definition of \(x_t\) as a global state and f as a global mechanism. Partial orderings may suffice provided a local state transition depends only upon information provided in the local state; see e.g. [13].
- 6.
The decision mapping defines an agent’s strategy set, which is a standard primitive defining a game; see e.g. [6].
- 7.
Although we focus upon real-valued laws here because of the estimation of continuous real-valued signals, in general finite or even infinite state machines may also characterize conservation laws.
- 8.
Cf. Proposition 1 of [29] for a proof of this assertion.
- 9.
These reflect a sampling of the permissible values of \(\sigma \)—a more detailed analysis is relegated to future research.
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Zargham, M., Paruch, K., Shorish, J. (2020). Economic Games as Estimators. In: Pardalos, P., Kotsireas, I., Guo, Y., Knottenbelt, W. (eds) Mathematical Research for Blockchain Economy. Springer Proceedings in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-030-53356-4_8
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