Abstract
In this chapter, we build on the concept of a repeated game and introduce the notion of a multistage game . In both types of games, several antagonistic agents interact with each other over time. The difference is that, in a multistage game, there is a dynamic system whose state keeps changing: the controls chosen by the agents in the current period affect the system’s future. In contrast with repeated games, the agents’ payoffs in multistage games depend directly on the state of this system. Examples of such settings range from a microeconomic dynamic model of a fish biomass exploited by several agents to a macroeconomic interaction between the government and the business sector. In some multistage games, physically different decision-makers engage in simultaneous-move competition. In others, agents execute their actions sequentially rather than simultaneously. We also study hierarchical games, where a leader moves ahead of a follower. The chapter concludes with an example of memory-based strategies that can support Pareto-efficient outcomes.
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- 1.
In the sections where we deal with dynamic systems described by multiple state equations, we adopt a notation where vectors and matrices are in boldface style to distinguish them from scalars that are in regular style.
- 2.
In stochastic systems, some “controls” may come from nature and are thus independent of other players’ actions.
- 3.
We should also note that, when the time horizon is infinite, it is usually assumed that the system is stationary . That is, the reward and state transition functions do not depend explicitly on time t.
- 4.
This limit is known as Cesaro limit .
- 5.
In the most general formulation of the problem, the control constraint sets may depend on time and the current state, i.e., \(U_{j}(t,x)\subset \mathbb {R}^{m_{j}}\). Moreover, sometimes the state may be constrained to remain in a subset \(\mathbf {X}\subset \mathbb {R} ^{n}\). We avoid these complications here.
- 6.
For example, the anti-block braking systems (ABS) used in cars, the automatic landing systems of aircrafts, etc.
- 7.
Commitments (agreements, treaties, schedules, planning processes, etc.) may force the agents to use the open-loop control even if state observations are available. On the other hand, some state variables (like the quantity of fish biomass in a management model for an ocean fishery) cannot be easily observable. In such cases, the agents may try to establish feedback controls using proxy variables, e.g., fish prices on a particular market.
- 8.
An information structure of this type is known as piecewise open-loop control .
- 9.
- 10.
We note that expressing payoffs as functions of players’ strategies is necessary for a game definition in normal form.
- 11.
To simplify notations, from now on we will omit the superscript T and refer to \(\tilde {\mathbf {u}}_j\) instead of \(\tilde {\mathbf {u}}_j^T\) or \(\tilde {\mathbf {x}}\) instead of \(\tilde {\mathbf {x}}^T\).
- 12.
Also called adjoint vector. This terminology is borrowed from optimal control theory.
- 13.
If a feedback strategy pair \( \underline {{\boldsymbol {\sigma }}}(t,\mathbf {x})\) is continuous in t and its partial derivatives \(\frac {\partial }{\partial \mathbf {x}} \underline {{\boldsymbol {\sigma }}}(t,\mathbf {x})\) exist and are continuous, then it is possible to characterize a feedback-Nash equilibrium through a coupled maximum principle (see Haurie et al. 2012).
- 14.
For notational simplicity, we still use Jj to designate this game’s normal form payoffs.
- 15.
In a stochastic context, perhaps counterintuitively, certain multistage ( supermodular) games defined on lattices admit feedback equilibria which can be established via a fixed-point theorem due to Tarski. See Haurie et al. (2012) and the references provided there.
- 16.
- 17.
This notation helps generalize our results. They would formally be unchanged if there were m > 2 players. In that case, − j would refer to the m − 1 opponents of Player j.
- 18.
We note that the functions \(W_{j}^*(\cdots )\) and \(W_{-j}^*(\cdots )\) are continuation payoffs . Compare Sect. 6.1.
- 19.
Recall that an equilibrium is a fixed point of a best-reply function, and that a fixed point requires some regularity to exist.
- 20.
The satisfaction of these conditions guarantees that such an equilibrium is feedback-Nash, or Markovian, equilibrium .
- 21.
- 22.
If it were possible to show that, at every stage, the local games are diagonally strictly concave (see e.g., Krawczyk and Tidball 2006), then one can guarantee that a unique equilibrium exists \( \underline {{\sigma }}(t,\mathbf {x})\equiv (\sigma _{j}(t, \mathbf {x}(t)),\sigma _{-j}(t,\mathbf {x}(t)))\,.{}\) However, it turns out that diagonal strict concavity for a game at t does not generally imply that the game at t − 1 possesses this feature.
- 23.
- 24.
- 25.
Any linear growth model in which capital expands proportionally to the growth coefficient a is called an AK model.
- 26.
Alternatively, we could postulate that the cost of adjusting output in the subsequent period is infinite.
- 27.
The announced strategy should be implemented. However, the leader could deviate from that strategy.
- 28.
That is, with a finite number of states. While games described by a state equation are typically infinite, matrix games are always finite.
- 29.
Source DeFreitas and Marshall (1998).
- 30.
In fact, the coefficients \(\underline {c}\), \( \underline {w}\), and Γ will all depend on α. However, to simplify notation, we will keep these symbols nonindexed.
- 31.
Where all \({ \underline {c}}^{\alpha },{ \underline {w}}^{\alpha },\Gamma _{c},\Gamma _{w}\) depend on β and α; see pages 203 and 206.
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Krawczyk, J.B., Petkov, V. (2018). Multistage Games. In: Başar, T., Zaccour, G. (eds) Handbook of Dynamic Game Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-44374-4_3
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