Matrix Games With Value 0 Continuous Function Linear Combination of Matrix Games

Game Theory

Chris Tsokos , Rebecca Wooten , in The Joy of Finite Mathematics, 2016

Problems

12.4.1

Find the expected payoff of the matrix game given by

$\begin{array}{ccc}\hfill \hfill & \hfill \hfill & \hfill {\mathit{P}}_{\mathbf{2}}\hfill \\ \hfill \hfill & \hfill & \begin{array}{ll}{\mathit{\gamma }}_{\mathbf{1}}\hfill & {\mathit{\gamma }}_{\mathbf{2}}\hfill \end{array}\hfill \\ {\mathit{P}}_{\mathbf{1}}\hfill & \hfill \begin{array}{c}\hfill {\mathit{\beta }}_{\mathbf{1}}\hfill \\ \hfill {\mathit{\beta }}_{\mathbf{2}}\hfill \end{array}\hfill & \hfill \left[\begin{array}{cc}\hfill \mathbf{-}\mathbf{6}\hfill & \hfill \mathbf{12}\\ \hfill \mathbf{4}& \hfill \mathbf{-}\mathbf{10}\hfill \end{array}\right]\hfill \end{array}$

If Player P ₁ chooses strategy β ₁ 60% of the time and Player P ₂ selects strategy γ ₂ 70% of the time. Which player does the game favor?

12.4.2

Determine the expected payoff for the game

$\begin{array}{cc}\hfill \hfill & \hfill {\mathit{P}}_{\mathbf{2}}\hfill \\ \hfill {\mathit{P}}_{\mathbf{1}}\hfill & \hfill \left[\begin{array}{rrr}\hfill \mathbf{10}\hfill & \hfill \mathbf{-}\mathbf{6}\hfill & \hfill \mathbf{2}\hfill \\ \hfill \mathbf{-}\mathbf{12}\hfill & \hfill \mathbf{8}\hfill & \hfill \mathbf{4}\hfill \\ \hfill \mathbf{16}\hfill & \hfill \mathbf{-}\mathbf{14}\hfill & \hfill \mathbf{-}\mathbf{8}\hfill \end{array}\right]\hfill \end{array}$

for the strategy probabilities [1/3 1/3 1/3] and $\left[\begin{array}{c}\hfill \raisebox{1ex}{$\mathbf{1}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{4}$}\right.\hfill \\ \hfill \raisebox{1ex}{$\mathbf{1}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{4}$}\right.\hfill \\ \hfill \raisebox{1ex}{$\mathbf{1}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{2}$}\right.\hfill \end{array}\right]$ for Players P ₁ and P ₂ , respectively.

12.4.3

Find the optimal strategies for Players P ₁ and P ₂ , and the value of the following games:

$\begin{array}{cc}\hfill \hfill & \hfill {\mathit{P}}_{\mathbf{2}}\hfill \\ \hfill {\mathit{P}}_{\mathbf{1}}\hfill & \hfill \left[\begin{array}{cc}\hfill \mathbf{-}\mathbf{4}\hfill & \hfill \mathbf{6}\\ \hfill \mathbf{8}& \hfill \mathbf{-}\mathbf{2}\hfill \end{array}\right]\hfill \end{array}\mathrm{and}\begin{array}{cc}\hfill \hfill & \hfill {\mathit{P}}_{\mathbf{2}}\hfill \\ \hfill {\mathit{P}}_{\mathbf{1}}\hfill & \hfill \left[\begin{array}{cc}\hfill \mathbf{10}& \hfill \mathbf{-}\mathbf{12}\hfill \\ \hfill \mathbf{-}\mathbf{16}\hfill & \hfill \mathbf{14}\end{array}\right]\hfill \end{array}$

12.4.4

In the game of matching pennies, consider the following situation: If Player P ₁ turns up T and P ₂ calls T , P ₁ pays P ₂ $5. However, if Player P ₂ calls H, then P ₂ must pay P ₁ $8. Now if P ₁ turns up H and P ₂ calls T , then P ₂ must pay P ₁ $10, but if P ₂ calls H, then P ₁ must pay P ₂ $12.

(a)

Obtain the matrix of the game.

(b)

Determine the optimal strategies for Players P ₁ and P ₂ .

(c)

Find the value of the game.

(d)

Does the game favor Player P ₁ ?

12.4.5

Find the optimal strategies for Players P ₁ and P ₂ and the value of the game

$\begin{array}{cc}\hfill \hfill & \hfill {\mathit{P}}_{\mathbf{2}}\hfill \\ \hfill {\mathit{P}}_{\mathbf{1}}\hfill & \hfill \left[\begin{array}{ccc}\hfill \mathbf{3}\hfill & \hfill \mathbf{7}\hfill & \hfill \mathbf{1}\hfill \\ \hfill \mathbf{2}\hfill & \hfill \mathbf{6}\hfill & \hfill \mathbf{8}\hfill \\ \hfill \mathbf{1}\hfill & \hfill \mathbf{3}\hfill & \hfill \mathbf{4}\hfill \end{array}\right]\hfill \end{array}$

12.4.6

Determine the optimal strategies for Players P ₁ and P ₂ and the value of the following game, if possible:

$\begin{array}{cc}\hfill \hfill & \hfill {\mathit{P}}_{\mathbf{2}}\hfill \\ \hfill {\mathit{P}}_{\mathbf{1}}\hfill & \hfill \left[\begin{array}{ccccc}\hfill \mathbf{3}& \hfill \mathbf{4}\hfill & \hfill \mathbf{-}\mathbf{4}\hfill & \hfill \mathbf{6}\hfill & \hfill \mathbf{0}\\ \hfill \mathbf{-}\mathbf{3}\hfill & \hfill \mathbf{5}\hfill & \hfill \mathbf{-}\mathbf{2}\hfill & \hfill \mathbf{7}\hfill & \hfill \mathbf{10}\hfill \end{array}\right]\hfill \end{array}$

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Evolutionary games: Natural selection of strategies

Irina Kareva , Georgy Karev , in Modeling Evolution of Heterogenous Populations, 2020

13.8 Discussion

In this chapter, we studied the process of natural selection of mixed strategies in classical 2x2 matrix games. The current state of the strategy selection process is described by the probability distribution of parameter α, which represents the (heritable) probability of an individual playing one of two strategies. This distribution can be considered as a distribution of mixed strategies in the game, and its dynamics are the main problem of interest.

The problem was completely solved using the hidden keystone variable (HKV) method; the distribution of parameter α at any time is defined by Eqs. (13.19)–(13.21). The dynamics of frequencies of pure strategies is described by Eq. (13.8), which generalizes a well-known replicator equation by Taylor and Jonker (1978).

These general results were applied to known 2   ×   2 matrix games. We were able to show that the dynamics of strategy distributions in the prisoner's dilemma (PD) and harmony (H) games essentially depend on the initial distribution of mixed strategies. In both cases the limit distribution is singular, that is, only a pure strategy can be selected over time from all possible mixed strategies, in agreement with known results.

Similarly, in the stag hunt (SH) game, only one pure strategy can be selected over time, but the initial population composition, and, more specifically, its mean value, has a critical impact on what strategy will finally be selected. In all of these cases, only a single pure strategy can be selected.

In contrast, in the hawk-dove (HD) game, not only the overall dynamics but also the shape of the final distribution of mixed strategies depends on the initial distribution. The final distribution is not singular, and any mixed strategy that was initially present in the population will be present in the final distribution. Another principal difference of the HD game from all other considered games is that KL divergence between current and initial distributions of strategies (information gain) tends to a finite value, while it tends to infinity for other games. We would like to emphasize that the HD game has clear biological interpretation and may be used to explain the persistence of species, whose members have potentially lethal attributes.

Interestingly the process of natural selection of strategies for all considered games obeys the dynamical principle of minimal information gain. It means that given the initial distribution and the value of the frequency of one of pure strategies, at any time, the current distribution of mixed strategies provides minimal information gain over all probability distributions. Formally, we can postulate this principle (as Kullback and Jaynes did in statistics and statistical physics accordingly) and then construct a solution to the model by solving a corresponding variational problem. What is important is that now we know for certain that this way we obtain the distribution that exactly coincides with the solution to the model. Hence the principle of minimal information gain is the underlying optimization principle, whose "invisible hand" governs the process of natural selection of strategies in these games.

In the next chapter, we describe a mathematical framework for analyzing natural selection not just between strategies, but between games. We show that the distribution of games changes over time due to natural selection. We also investigate the question of mutual invasibility of games with respect to different strategies and different initial population composition. Finally, we discuss the applicability of the developed framework to understanding games that cancers play.

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Inherently Parallel Algorithms in Feasibility and Optimization and their Applications

L.M. Bregman , I.N. Fokin , in Studies in Computational Mathematics, 2001

2 NASH EQUILIBRIUM AND EQUIVALENT MATRIX GAME

Let X_i be the set of pure strategies of player i in game Γ. We denote ∏ _{i  ∈ I} X _i by X. For the resource allocation game,

Let s_i be a mixed strategy for player i,i.e s_i is a probability distribution on Xⁱ :

Let S_i be the set of mixed strategies for player i, and S  :=   ∏ _{i  ∈ I} S _i the set of randomized- strategy profiles. For s ∈ S, the expected value of the payoff function for player i is

However, taking into account that player interactions are pairwise, we can represent π_i (s) in a simpler form:

Where

${\pi }_{ij}\left(s_i,s_j\right):={\displaystyle \sum _{x_i\in X_i,x_j\in X_j}{\pi }_{ij}\left(x_i,x_j\right)}{s}_i\left(x_i\right)s_j\left(x_j\right)\text{.}$

For any t_i ∈ S_i , we let (s_-i, t_i ) denote the randomized-strategy profile in which the i-th component is t_i and all other components are as in s. Thus,

$\pi \left({S_{-}}_i,t_i\right)={\displaystyle \sum _{x\in X}\left({\displaystyle \prod _{j\in I\backslash i}{S}_j}\left(x_j\right)\right)}{t}_i\left(x_i\right){\pi }_i\left(x\right)\text{,}$

or

(4) $\pi \left({S_{-}}_i,t_i\right)={\displaystyle \sum _{j\in I}}{\displaystyle \sum _{x_i\in X_i,x_j\in X_j}}{\pi }_{ij}\left(x_i,x_j\right)t_i\left(x_i\right)s_j\left(x_j\right)\text{.}$

Definition 2.1

We say that s ∈ S is a Nash equilibrium if

${\pi }_i\left(s\right)\ge {\pi }_i\left(s_{-i},t_i\right),\mathrm{for\; each}i\in I,t_i\in S_i\text{.}$

Now we construct a fair matrix game ∆ which is equivalent to game Γ, that is, each randomized-strategy profile in Γ corresponds to some mixed strategy for a player in ∆ and vice versa. The strategy corresponding to the Nash equilibrium is an optimal mixed strategy in ∆. Since game ∆ is fair (i.e. its value is 0 and the sets of optimal strategies for the first and second player are the same), an optimal mixed strategy can be found by the solution of some linear feasibility problem.

Construct |X _i |   ×   |X _j | -matrices H_ij an element in the row corresponding to the strategy x_i ∈ X_i and in the column corresponding to the strategy x_j ∈ X_j is defined as H_ij [x_i , x_j ] := π_ij (x_i , x_j ). H_ii is defined as |X_i |   ×   |X _j | -matrix of zeros. It is clear that H _ij   =   − H _ji ^T .

Then we consider a matrix H consisting of blocks H_ij

$H:=\left(\begin{array}{cccc}\hfill H_{{}^{{}^{11}}}\hfill & \hfill H_{12}\hfill & \hfill \dots \hfill & \hfill H_{1\left|I\right|}\hfill \\ \hfill \cdot \hfill & \hfill \cdot \hfill & \hfill \dots \hfill & \hfill \cdot \hfill \\ \hfill \cdot \hfill & \hfill \cdot \hfill & \hfill \dots \hfill & \hfill \cdot \hfill \\ \hfill H_{\left|I\right|1}\hfill & \hfill H_{\left|I\right|2}\hfill & \hfill \dots \hfill & \hfill H_{\left|I\right|\left|I\right|\text{.}}\hfill \end{array}\right)$

Matrix H is a square matrix of order ∑ _{i  ∈ I} |X _i | It is a skew-symmetric matrix, that is, H  =   -H^T .

Consider a matrix B consisting of zeros and ones. Each row of B corresponds to some x ∈ X, that is, B has $m:={\displaystyle {\prod }_{i\in I}}\left|X_i\right|$ rows. The columns of B are divided into |I| blocks. The i-th block B_i corresponds to set X_i and consists of |X_i | columns, each column corresponding to some x_i ∈ X_i . Matrix B has |I| ones in each row: one 1 in each block, namely, if some row corresponds to x  =   (x_i ),i ∈ I, then it has 1 in the i-th block in the column corresponding to strategy x_i .

Let us consider now a matrix game ∆ with (m   ×   m)-matrix

(5) $A=BH{B}^{\mathrm{T.}}$

It is clear that matrix A is skew-symmetric. It means that game ∆ is fair, its value is 0 and the sets of optimal strategies for the first and second player are the same.

Lemma 2.2

If a vector y is a mixed strategy for the second player in ∆, then s  = B^T y is a randomized-strategy profile in Γ. The reverse: if s is a randomized-strategy profile in Γ, then there exists a vector y satisfying B^T y   =   s which is a mixed strategy for the second player in ∆.

Proof. Let y be a mixed strategy for the second player in ∆, and s  = B^Ty. Then s  =   (s_i ), i ∈ I, where s_i   = B _i ^T y. Since B _i has one unity in each row, and ∑ _{x  ∈ X} y(x)   =   1, we have ex ∑ _{xi  ∈ Xi} s _i (x _i )   =   1, for all i ∈ I. It is clear that s_i   ≥   0, for all i and xⁱ because B_i and y are non-negative. So s_i is a mixed strategy for player i in Γ, and s is a randomized- strategy profile.

Let s=  (s_i ), i ∈ I, be a randomized-strategy profile in Γ.

Consider a vector y with components y(x), x ∈ X, defined as

$y\left(x\right):={\displaystyle \prod _{i\in I}{s}_i}\left(x_i\right)\text{.}$

It is clear that y is a probability vector, that is, y is a mixed strategy for the second player in ∆. Show that B _i ^T y  = s _i , for all i ∈ I.

Let t be a component of vector B _i ^T y corresponding to some z ∈ X_i. Then

$t={\displaystyle \sum _{\left\{x\in X:x_i=z\right\}}{\displaystyle \prod _{j\in I}{s}_j\left(x_j\right)=}}{s}_i\left(z\right){\displaystyle \prod _{j\ne i}{\displaystyle \sum _{x_j\in X_j}{s}_j\left(x_j\right)}}\text{.}$

Since ${\displaystyle \sum {{}_{xj\in Xj}}^{sj(xj)}=1},$ we have B _i ^T y  = s _i , that is, B^Ty = s ∎.

Obviously, the lemma's statements also relate to the first player strategies.

Theorem 2.3

Game ∆ is equivalent to Γ, i.e. if y is an optimal strategy for the second player in ∆, then s  = B^Ty is a Nash equilibrium in Γ, and if s ∈ S is a Nash equilibrium in Γ, then each y satisfying s  = B^Ty is an optimal strategy for the second player in ∆.

Proof. Let y be an optimal strategy for the second player in ∆. Since ∆ is fair, this means that

(6) ${\displaystyle {\sum _{y\ge 0.}^{Ay\le 0,}}_{x\in X}y\left(x\right)}=1\text{,}$

Let s=B^Ty, s  =   (s_i ),i ∈ I, is a mixed strategy for the i-th player.Let t_i be another mixed strategy for player i, and u   =   (s _{- i} ,t_i ).

By lemma 2.2, there exists a mixed strategy for the first player z such that zB  = u. By (6), we have

$\mathrm{zAy}\le 0\text{,}$

or taking into account representation (5),

$\mathrm{uHs}\le 0\text{.}$

The latter inequality is equivalent to

${\displaystyle \sum _{k\in I,j\in I}{u}_k{H}_{kj}{s}_j\le 0\text{.}}$

By definition of H_kj ,

$u_k{H}_{kj}{s}_j={\displaystyle \sum _{x_k\in X_k,x_j\in X_j}\pi \left(x_k,x_j\right)u_k\left(x_k\right)s_j\left(x_j\right)}$

For k  ≠ i,

${\displaystyle \sum _{j\in I}{u}_k{H}_{kj}{s}_j}={\displaystyle \sum _{j\in I}{s}_k{H}_{kj}{s}_j}={\displaystyle \sum _{j\in I}}{\displaystyle \sum _{x_k\in X_k,x_j\in X_j}\pi \left(x_k,x_j\right)s_k\left(x_k\right)s_j\left(x_j\right)}={\pi }_k\left(s\right)\text{,}$

and for k = i by (4),

${\displaystyle \sum _{j\in I}{u}_i{H}_{ij}{s}_j}={\displaystyle \sum _{j\in I}{t}_i{H}_{ij}{s}_j}={\displaystyle \sum _{j\in I}}{\displaystyle \sum _{x_i\in X_i,x_j\in X_j}\pi \left(x_i,x_j\right)t_i\left(x_i\right)s_j\left(x_j\right)}={\pi }_i\left(s_{-i},t_i\right)\text{.}$

These relations imply that

${\displaystyle \sum _{k\ne i}{\pi }_k\left(s\right)}+{\pi }_i\left(s_{-i},t_i\right)\le 0\text{,}$

and since Γ is a zero sum game, i.e. ∑ _{k  ∈ I} πk(s)   =   0, we have

${\pi }_i\left(s_{-i},t_i\right)\le {\pi }_i\left(s\right)\text{.}$

Since i ∈ I and t_i ∈ S_i was chosen arbitrarily, this means that s is a Nash equilibrium in Γ.

Let now s be a Nash equilibrium in Γ. Take x ∈ X: x  =   (x_i ), i ∈ I. Let u_i be the mixed strategy for the i-th player which chooses the strategy x_i with the probability 1, and u := (u_i ,), i ∈ I. Then u  = b_x , where b_x is the row of matrix B corresponding to the strategy profile x. For each i ∈ I, we have

${\pi }_i\left(s_{-i},u_i\right)\le {\pi }_i\left(s\right)\text{.}$

Summing these inequalities, we obtain

${\displaystyle \sum _{i\in I}{\pi }_i\left(s_{-i},u_i\right)\le {\displaystyle \sum _{i\in I}{\pi }_i\left(s\right)}=0\text{.}}$

Using (4), we have

$\mathrm{uHs}\le 0\text{.}$

Let y be a mixed strategy for the second player in ∆ such that s =  B^Ty (such y exists by lemma 2.2). Then b_xHB^Ty   ≤   0. Since x was chosen arbitrarily, we have

${\mathrm{BHB}}^{T}y\le 0\text{,}$

that is, y is an optimal strategy for the second player in ∆. ∎

Thus, the problem of finding a Nash equilibrium in Γ has been reduced to the solution of a system of linear inequalities (6). However, the number of variables and constraints in (6) is huge because the order of matrix A is

${\displaystyle \prod _{i\in I}\left|X_i\right|}={\displaystyle \prod _{i\in I}\left(\begin{array}{l}{k}_i+\left|N\right|-1\\ \left|N\right|-1\end{array}\right)}\text{.}$

We need to exploit specific properties of matrix A to construct an efficient method for the solution of system (6). The basic property is that the rank of matrix A is significantly smaller than its order, and we can represent this matrix as a product of matrices with relatively small size.

Representation (5) gives some factorization for matrix A, and we can see that the rank of A is not greater than the number of rows in H which is equal to the sum of the numbers of pure strategies of all the players. This is significantly smaller than the number of rows (or columns) in A which is equal to the product of these numbers, but it is still a large number. We construct a factorization for matrix H which shows that in fact the rank of matrix A is much smaller than that of H. This factorization makes it possible to construct an efficient method for the solution of (6) which operates with arrays of relatively small size.

We begin with a factorization of matrices H_ij. The factorization reflects the additive form of the elements of these matrices expressed by (1).

Let us introduce (K_i   +   1)   ×   (K_j   +   1)- matrices p_ijv , v ∈ N, whose elements are payoffs for player i from player j on terrain v, i.e. p_ijv [k_i ,k_j ] := π_ijv (k_i ,k_j ), 0   ≤ k_i   ≤ K_i , 0   ≤ k_j   ≤   K_j. Construct a block-angular matrix G_ij , which is composed of blocks p _ijv:

${G_i}_j:=\left(\begin{array}{cccc}\hfill p_{ij1}\hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill p_{ij2}\hfill & \hfill \hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill \cdots \hfill & \hfill \hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill p_{ij\left|N\right|}\hfill \end{array}\right)\text{.}$

G_ij is an (|N|(K_i   +   1) x |N|(K_j   +   1))-matrix.

Let us consider zero-one matrices R_iv , i ∈ I,v ∈ N. Each row of R_iv corresponds to some pure strategy of player i. The number of columns in this matrix is equal to K_i   +   1. The element in the row corresponding to strategy x_i and in column k (0   ≤ k  ≤ K_i ) is equal to one, if player i allocates k units of the resource to terrain v in his strategy xⁱ . R_iv are (|X_i |   ×   (K_i   +   1))-matrices. Compose a matrix R_i from matrices R_iv merging their columns for all v ∈ N: R_i := (R_i1 , R_i2 ,…, R_i|N| ), R_i is an (|X_i |   ×   |N|(K_i   +   1))-matrix.

Then matrix H_ij can be represented in the following form:

and matrix H can be represented in the form H  = RGR^T , where matrices R and G are composed of R_i and G_ij :

$G:=\left(\begin{array}{cccc}\hfill G_{11}\hfill & \hfill G_{12}\hfill & \hfill \dots \hfill & \hfill G_{1\left|I\right|}\hfill \\ \hfill \cdot \hfill & \hfill \hfill & \hfill \dots \hfill & \hfill \cdot \hfill \\ \hfill \cdot \hfill & \hfill \hfill & \hfill \cdots \hfill & \hfill \cdot \hfill \\ \hfill G_{\left|I\right|1}\hfill & \hfill G_{\left|I\right|2}\hfill & \hfill \dots \hfill & \hfill G_{\left|I\right|\left|I\right|}\hfill \end{array}\right)\text{.}$

R is a (∑ _{i  ∈ I} |X _i |   ×   |N|(∑ _{i  ∈ I} K _i   +   |I|)) -matrix, and G is a square matrix of the order |N|(∑ _{i  ∈ I} K _i   +   |I|).

Hence matrix A can be represented as a product of five matrices: A  = BRGR^TB^T . The rank of A is not greater than the minimal rank of these matrices. Therefore the rank of A is significantly lower than its order. The order of A is

and the rank of A is less than or equal to the rank of matrix G, that is,

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Game Theory

Guillermo Owen , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

II Two-Person Zero-Sum Games

It is for dual, or two-person zero-sum games, that the most satisfactory theory has been developed. For these games, the sum of the two players' payoffs is always zero; hence, a single number (the amount won by the first player, and therefore lost by the second) determines the payoff. In the finite case, the normal form for such a game is a matrix (as inFig. 2 ), with each row representing one of Player 1's strategies, and each column, one of player 2's strategies. Such games are called matrix games.

Consider the matrix game shown inFig. 3. It can be seen that, if Player 1 chooses row E, he can be certain of winning at least 2 units. On the other hand, with row D, he might win as little as 1 unit, and, with row F, he might even lose 3 units, depending on what Player 2 does. Thus row E is his maximin strategy (it maximizes his minimum winnings). In a similar way, column A represents Player 2's minimax strategy; i.e., the maximum entry, 2, in this column, is the minimum of the column maxima.

FIGURE 3. A game with optimal pure strategies.

InFig. 3, the maximin and minimax are both equal to 2. By choosing row E, Player 1 is sure of winning at least 2; by choosing column A, Player 2 is sure of losing no more than 2. It is then suggested that both players should choose the maximin/minimax strategies, which are known as optimal strategies.

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Public Choice

Hartmut Kliemt , in Philosophy of Economics, 2012

1.2 Public choice as a game

The implications of the homo oeconomicus model began to be spelled out in more detail in non-co-operative game theory almost exactly when Public Choice started its rise. In all likelihood this is pure coincidence, yet one might argue that it makes very good methodological sense. There is one game of life with one type of rational individual populating that world. Specific results derive from the specific rules of partial or lower order specific games. Specific games like markets or voting in politics etc. are merely abstracted from the broader context to make them analytically tractable. But besides this the same methods can be applied to the various kinds of games. In short, homo oeconomicus plays all the games people play (alluding, of course, to [Maital and Maital, 1984]).

Though using the game metaphor may not seem too exciting it has stronger implications than often recognised. For, once an interaction is conceptualised as a game in non-co-operative game theoretic terms it is clear that the results of the interaction must be emergent rather than chosen. Single-handedly, players cannot "choose" results of a game. That this is so is the whole point of a strategic game conceptualisation of an interaction. Players in a game have full control over their individual moves but, except for special cases to which we tend to refer as "games against nature", no player has full control over results. Players therefore cannot "choose" outcomes of proper strategic games at least not in the non-metaphorical or narrow sense of choosing an option. In short, in the non-cooperative conceptualization of a strategic game the choice of a result is not among the options of choice — though choices of options lead to results.

For a most simple illustration consider a 2 × 2 matrix game. As in particular James M. Buchanan in his use of the metaphor of the 2 × 2 matrix game for "collective choice" has always insisted, the two players can not properly speaking "choose" a result. They can either choose a column — as column player — or a row — as row player. Each can choose one of the two moves open to each of them but none can choose one of the cells (see in particular [ Buchanan, 1975/1996] and his earlier criticisms of the Social Choice paradigm as reprinted in [Buchanan, 1999]). None of the players can choose unilaterally a cell. This is impossible unless the other player were just a puppet on the strings of the choosing actor. Then the actor merely would play against "nature" rather than a strategic game against a co-player who herself is an independent centre of choice making.

In a strategic game individuals can, of course, rank collective results as might emerge for instance by means of a personal social welfare function (see below part 2.2. and also the discussion of the so called liberal paradox in the value ordering [Sen, 1996] as opposed to the game form conceptualisation as in [Buchanan, 1975/1996; Gaertner et al. , 1992; Sugden, 1994]). The insight that even in the most simple case of a 2 × 2 matrix game the results of a play of the game are not chosen but are necessarily emergent obviously extends to games with any number of players, moves, and strategies. Any conceptualisation of a social interaction in terms of non-co-operative game theory will imply it. The framework of non-co-operative game theory explicitly models all moves and thereby all causal influences of individuals on each other and their environment. It forms the most detailed and basic conceptual scheme for representing any form of social interaction.

Since public choice is a social interaction it is clear that a non-co-operative game account of it should be regarded as fundamental within Public Choice. As far as this is concerned it seems significant that Buchanan has always endorsed (non-cooperative) game theory as a conceptual tool of Public Choice (see in particular [Buchanan, 2001]). One should, however, bear in mind that classical or, to use Ken Binmore's (see [Binmore, 1987/88]) apt term, "eductive" (non-co-operative) game theory is not a behavioural theory at all but rather a "theory of reasoning about knowledge" (in the sense of [Fagin et al., 1995]). The view that game theory "predicts" or "explains" what we observe in behavioural terms is, if we take these terms literally rather than merely metaphorically, quite far-fetched.

Buchanan's Constitutional Political Economy approach shares — without calling it by that name — to some extent the outlook of eductive game theory. But as he is quite well aware this stands in an uneasy relationship to explanatory science more narrowly and traditionally conceived (see [Buchanan, 1982]). Those who see Public Choice rather as an empirical behavioural science should be quite unhappy with Public Choice adopting the methods of "a logic of choice making". As far as explaining human behaviour — in a covering law sense of explanation (see [Hempel and Oppenheim, 1948]) — is concerned eductive theory has not much value. There are no behavioural laws or anything like such laws in it (or the latter must be added, see [Hempel, 1965/1970, essay 9]). The contribution of eductive game theory to forming an explanatory theory is exclusively that of a modelling tool or a language in which substantive theories can be precisely expressed ("Rational Choice Modelling" rather than "Rational Choice Theory", as defined in [Güth and Kliemt, 2007]). As opposed to that classical conceptualisations of Public Choice as formulated in Political Philosophy had some, albeit sometimes very moderate, substantive content.

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