Abstract
This chapter provides a general introduction to the theory of games, as a prelude to other chapters in this Handbook of Dynamic Game Theory which discuss in depth various aspects of dynamic and differential games. The present chapter describes in general terms what game theory is, its historical origins, general formulation (concentrating primarily on static games), various solution concepts, and some key results (again primarily for static games). The conceptual framework laid out here sets the stage for dynamic games covered by other chapters in the Handbook.
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Notes
- 1.
It is also possible to define games where the Players set is not finite. This chapter will not discuss such games, known generally as mean field games when some structure is imposed on the way other players’ decision variables enter into the objective function of a particular (generic) player. Another chapter in this Handbook will address primarily such games.
- 2.
Upper and lower values are defined in more general terms using infimum (\(\inf \)) and supremum (\(\sup \)) replacing minimum and maximum, respectively, to account for the facts that minima and maxima may not exist. When the action sets are finite, however, the latter always exist.
- 3.
Here the positivity requirement on each αi is uniform for all x−i, that is, there exists a constant 𝜖 > 0 such that \(\alpha _i(x_{-i}) > \epsilon \; \forall \, x_{-i}\in X_{-i},\; i\in {\mathcal {N}}\).
- 4.
In ZS matrix games, under the convention adopted in this chapter regarding the roles of the players, we say that a row strictly dominates another row if the difference between the two vectors (first one minus the second one) has all negative entries. Likewise, a column strictly dominates another column if the difference has all positive entries.
- 5.
The notation 1m below stands for the m-dimensional column vector whose entries are all 1s.
- 6.
Even though the discussion in this section uses the framework of N-player noncooperative games with NE as the solution concept, it applies as a special case to two-player zero-sum games, by taking J1 = −J2 and noting that in this case NE becomes SPE.
- 7.
Using the earlier convention, the notation γ−i stands for the collection of all players’ strategies, except the i’th one.
- 8.
An NE is said to be admissible if there is no other NE which yields better outcome for all players.
- 9.
Selten’s construction and approach also apply to static games of the types discussed heretofore, where slight perturbations are made in the entries of the matrices, instead of at information sets.
- 10.
A game is one with perfect recall if all players recall their past moves – a concept that applies to games in extensive form.
- 11.
As introduced in the previous section, behavioral strategy is a mixed strategy for each information set of a player (in a dynamic game in extensive form). When the context is static games, it is identical with mixed strategy.
- 12.
This is also called “trembling hand equilibrium,” as the process of erring at each information set is reminiscent of a “trembling hand” making unintended choices with small probability. Here, as k →∞, this probability of unintended plays converges to zero.
- 13.
Mi is the set of all pure strategies of Player 1, with corresponding labeling of positive integers.
- 14.
In this asymmetric decision-making setting, we will refer to Player 1 as “she” and Player 2 as “he.”
- 15.
Of course, the “strategy” here could also be viewed as an “action” if what we have is a static game, but since we are dealing with normal forms here (which could have an underlying extensive form description) we will use the term “strategy” throughout, to be denoted by γi for Pi, and the cost to Pi will be denoted by Ji.
- 16.
This fixed point theorem says that if S is a compact subset of \(\mathbb {R}^n\), and f is an upper semicontinuous function which assigns to each x ∈ S a closed and convex subset of S, then there exists x ∈ S such that x ∈ f(x).
- 17.
Brouwer’s theorem says that a continuous mapping, f, of a closed, bounded, convex subset, S, of a finite-dimensional space into itself has a fixed point.
- 18.
Ti is known as the reaction function (or response function) of Player i to other players’ actions.
- 19.
Here, existence of SPE is a direct consequence of Theorem 4. By strict convexity and strict concavity, there can be no SPE outside the class of pure strategies, and uniqueness follows from the ordered interchangeability property of multiple SPs, in view of strict convexity/concavity (Başar and Olsder 1999).
- 20.
The underlying idea of the proof is to make the kernels Li discrete so as to obtain an N-person matrix game that suitably approximates the original game in the sense that an MSNE of the latter (which exists by Nash’s theorem) is arbitrarily close to a mixed equilibrium solution of the former. Compactness of the action spaces ensures that a limit to the sequence of solutions obtained for approximating finite matrix games exists.
- 21.
The qualifier genuine is used here to stress the point that mixed strategies in this statement are not pure strategies (even though pure strategies are indeed special types of mixed strategies, with all probability weight concentrated on one point).
- 22.
This follows from Banach’s contraction mapping theorem. If T maps a normed space X into itself, it is a contraction if there exists α ∈ [0, 1) such that ∥T(x) − T(y)∥≤ α∥x − y∥, ∀x, y ∈ X.
- 23.
In retrospect, this should not be surprising since for the special case of ZSGs (without pure-strategy saddle points), we have already seen that the minimizer could further decrease her guaranteed expected cost by playing a mixed strategy; here, however, it holds even if J1≢ − J2.
- 24.
This one corresponds to (1.22).
- 25.
For a convex-concave quadratic game, the upper value will not be bounded if, and only if, there exists a \(v\in \mathbb {R}^{m_2}\) such that \(v'R_{22}^2v =0\) while \(v'r^1_2 \neq 0\). A similar result also applies to the lower value.
- 26.
This result may fail to hold true for team problems with strictly convex but nondifferentiable kernels.
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Başar, T. (2018). Introduction to the Theory of Games. In: Başar, T., Zaccour, G. (eds) Handbook of Dynamic Game Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-44374-4_1
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