Overview
--------
-Dunshire is a CVXOPT-based library for solving linear (cone)
-games. The notion of a cone game was introduced by Gowda[1] and
-extended to asymmetric cones in my thesis[2].
+Dunshire is a `CVXOPT <http://cvxopt.org/>`_-based library for solving
+linear (cone) games. The notion of a symmetric linear (cone) game was
+introduced by Gowda and Ravindran [GowdaRav]_, and extended by
+Orlitzky to asymmetric cones with two interior points.
+
+The state-of-the-art is that only symmetric games can be solved
+efficiently, and thus the linear games supported by Dunshire are a
+bastard of the two: the cones are symmetric, but the players get to
+choose two interior points.
Requirements
------------
distributions. Dunshire is targeted at python-3.x, but python-2.x will
probably work too.
-[1] M. S. Gowda and G. Ravindran. On the game-theoretic value of a
- linear transformation relative to a self-dual cone. Linear Algebra
- and its Applications, 469:440-463, 2015.
+Background
+----------
+
+A linear game is a generalization of a two-person zero-sum matrix
+game~\cite{karlin, parthasarathy-raghavan,
+von_neumann-morgenstern}. Classically, such a game involves a matrix
+:math:`A \in \mathbb{R}^{n \times n}` and a set (the unit simplex) of
+"strategies" :math:`\Delta = \operatorname{conv}\left(
+\left\lbrace e_{1}, e_{2}, \ldots, e_{n} \right\} \right)` from which two
+players are free to choose. If the players choose :math:`x,y \in \Delta`
+respectively, then the game is played by evaluating :math:`y^{T}Ax` as the
+"payoff" to the first player. His payoff comes from the second
+player, so that their total sums to zero.
+
+Each player will try to maximize his payoff in this scenario,
+or—what is equivalent—try to minimize the payoff of his
+opponent. In fact, the existence of optimal strategies is
+guaranteed [Karlin]_ for both players. The *value* of the
+matrix game :math:`A` is the payoff resulting from optimal play,
+
+.. math::
+ v\left(A\right)
+ =
+ \underset{x \in \Delta}{\max}\
+ \underset{y\in \Delta}{\min}
+ \left( y^{T}Ax \right)
+ =
+ \underset{y \in \Delta}{\min}\
+ \underset{x \in \Delta}{\max}
+ \left( y^{T}Ax \right).
+
+The payoff to the first player in this case is
+:math:`v\left(A\right)`. Corresponding to :math:`v\left(A\right)` is an
+optimal strategy pair :math:`\left(\bar{x}, \bar{y}\right) \in \Delta
+\times \Delta` such that
+
+.. math::
+ \bar{y}^{T}Ax
+ \le
+ v\left(A\right)
+ =
+ \bar{y}^{T}A\bar{x}
+ \le
+ y^{T}A\bar{x}
+ \text{ for all } \left(x,y\right) \in \Delta \times \Delta.
+
+The relationship between :math:`A`, :math:`\bar{x}`, :math:`\bar{y}`,
+and :math:`v\left(A\right)` has been studied extensively [Kaplansky]_
+[Raghavan]_. Gowda and Ravindran [GowdaRav]_ were motivated by these
+results to ask if the matrix :math:`A` can be replaced by a linear
+transformation :math:`L`, and whether or not the unit simplex
+:math:`\Delta` can be replaced by a more general set—a base of a
+symmetric cone. In fact, one can go all the way to asymmetric (but
+still proper) cones. But, since Dunshire can only handle the symmetric
+case, we will pretend from now on that our cones need to be
+symmetric. This simplifies some definitions.
+
+What is a linear game?
+----------------------
+
+In the classical setting, the interpretation of the strategies as
+probabilities results in a strategy set :math:`\Delta \subseteq
+\mathbb{R}^{n}_{+}` that is compact, convex, and does not contain the
+origin. Moreover, any nonzero :math:`x \in \mathbb{R}^{n}_{+}` is a
+unique positive multiple :math:`x = \lambda b` of some :math:`b \in
+\Delta`. Several existence and uniqueness results are predicated on
+those properties.
+
+**Definition.** Suppose that :math:`K` is a cone and :math:`B
+\subseteq K` does not contain the origin. If any nonzero :math:`x \in
+K` can be uniquely represented :math:`x = \lambda b` where
+:math:`\lambda > 0` and :math:`b \in B`, then :math:`B` is a *base*
+of :math:`K`.
+
+The set :math:`\Delta` is a compact convex base for the proper cone
+:math:`\mathbb{R}^{n}_{+}`. Every :math:`x \in \Delta` also has
+entries that sum to unity, which can be abbreviated by the condition
+:math:`\left\langle x,\mathbf{1} \right\rangle = 1` where
+:math:`\mathbf{1} = \left(1,1,\ldots,1\right)^{T}` happens to lie in
+the interior of :math:`\mathbb{R}^{n}_{+}`. In fact, the two
+conditions :math:`x \in \mathbb{R}^{n}_{+}` and :math:`\left\langle
+x,\mathbf{1} \right\rangle = 1` define :math:`\Delta`. This is no
+coincidence; whenever :math:`K` is a symmetric cone and :math:`e_{2}
+\in \operatorname{int}\left(K\right)`, the set :math:`\left\lbrace x
+\in K \ \middle|\ \left\langle x,e_{2} \right\rangle = 1
+\right\rbrace` is a compact convex base of :math:`K`. This motivates a
+generalization where :math:`\mathbb{R}^{n}_{+}` is replaced by a
+symmetric cone.
+
+**Definition.** Let :math:`V` be a finite-dimensional real Hilbert
+space. A *linear game* in :math:`V` is a tuple
+:math:`\left(L,K,e_{1},e_{2}\right)` where :math:`L : V \rightarrow V`
+is linear, the set :math:`K` is a symmetric cone in :math:`V`, and the
+points :math:`e_{1}` and :math:`e_{2}` belong to
+:math:`\operatorname{int}\left(K\right)`.
+
+References
+----------
+
+.. [GowdaRav] M. S. Gowda and G. Ravindran. On the game-theoretic
+ value of a linear transformation relative to a self-dual cone. Linear
+ Algebra and its Applications, 469:440-463, 2015.
+
+.. [Kaplansky] I. Kaplansky. A contribution to von Neumann’s theory of
+ games. Annals of Mathematics, 46(3):474-479, 1945.
-[2] You wish.
+.. [Karlin] S. Karlin. Mathematical Methods and Theory in Games,
+ Programming, and Economics. Addison-Wesley Publishing Company,
+ Reading, Massachusetts, 1959.
-[3] http://cvxopt.org/
+.. [Raghavan] T. Raghavan. Completely mixed games and M-matrices. Linear
+ Algebra and its Applications, 21:35-45, 1978.
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