Best Experienced Payoff Dynamics and Cooperation in the Centipede Game: Online Appendix

Transcription

1 Best Experienced Payoff Dynamics and Cooperation in the Centipede Game: Online Appendix William H Sandholm, Segismundo S Izquierdo, and Luis R Izquierdo February 8, 018 Contents I Exact and numerical calculation in Mathematica I1 Algebraic numbers and solutions to polynomial equations I Algorithms from computational algebra 3 I3 Numerical evaluation and precision tracking 3 II The BEP Centipedenb notebook 4 II1 Exact analysis 4 II Numerical analysis 5 II3 More on computation of approximate rest points and eigenvalues 6 III Dimension reduction for local stability analysis 7 IV Time until convergence under different test-set rules 10 V Analyses of repulsion from the backward induction state 13 V1 Test-two, stick-if-tie: 13 V Test-adjacent, stick-if-tie: 14 VI Formulas for BEP(τ, 1, β) dynamics in Centipede 15 VI1 Test-all 15 VI Test-two 16 VI3 Test-adjacent 17 VII Multinomial formulas for BEP(τ, κ, β) dynamics 18 Department of Economics, University of Wisconsin Department of Industrial Organization, Universidad de Valladolid Department of Civil Engineering, Universidad de Burgos

2 VIII Approximate components of interior rest points 0 VIII1 Test-all 1 VIII Test-two 5 VIII3 Test-adjacent 9 IX Approximate eigenvalues of DV(ξ ) 33 IX1 Test-all 33 IX Test-two 35 IX3 Test-adjacent 37 I Exact and numerical calculation in Mathematica In this section we describe the built-in Mathematica functions we use to prove exact (analytical) results and to obtain numerical evaluations of exact expressions I1 Algebraic numbers and solutions to polynomial equations To obtain our analytical results, we take advantage of Mathematica s ability to perform exact computations using algebraic numbers As described in Strzeboński (1996, 1997), Mathematica represents algebraic numbers using Root objects, with Root[poly, k] designating one of the roots of the minimal polynomial poly The index k is used to single out a particular root of poly, with the lowest indices referring to the real roots of poly in increasing order, and the higher indices referring to the complex roots in a more complicated way Root objects also contain a hidden third element that specifies an isolating set for the root, meaning a set containing the root of poly in question and no others The forms of isolating sets depend on whether roots are isolated using arbitraryprecision floating point methods or exact methods If Mathematica s default settings are used, then roots are isolated using arbitrary-precision floating point methods based on the Jenkins-Traub algorithm (Jenkins (1969), Jenkins and Traub (1970a,b)), the workhorse numerical algorithm for this purpose While in theory this algorithm always isolates all real and complex roots of poly in disjoint disks in the complex plane, flawless implementation of the algorithm is difficult; see Strzeboński (1997, p 649) If we instead use the setting SetOptions[Root,ExactRootIsolation->True] then Mathematica isolates roots using exact methods that is, methods that only use rational number calculations Real roots of polynomials are isolated in disjoint intervals using the Vincent-Akritas-Strzeboński method, which is based on Descartes rule of signs and a classic theorem of Vincent; see Akritas et al (1994) and Akritas (010) Complex roots are isolated in rectangles using the Collins and Krandick (199) method Exact roots of univariate polynomials (and much else) can be computed using the Mathematica function Reduce When computing the exact rest points of BEP dynamics, we apply Reduce to the output of the function GroebnerBasis, described next

3 I Algorithms from computational algebra The Mathematica function GroebnerBasis is an implementation of a proprietary variation of the algorithm of Buchberger (1965, 1970) 1 Choosing the option Method -> Buchberger causes Mathematica to use the original Buchberger algorithm, which runs considerably more slowly than the default algorithm; however, there was only one case in which the default algorithm produced a Gröbner basis and the Buchberger algorithm failed to terminate The Mathematica function CylindricalDecomposition implements the Collins (1975) cylindrical algebraic decomposition algorithm with various improvements If this function is run in its default mode, it makes use of arbitrary-precision arithmetic To force Mathematica to work with algebraic numbers, one uses the following settings: SetOptions[Root,ExactRootIsolation->True] SetSystemOptions[ InequalitySolvingOptions -> CADDefaultPrecision ->Infinity] Unfortunately, these settings cause CylindricalDecomposition to run extremely slowly, and in the case of BEP dynamics in Centipede it only generates a result in two-dimensional cases Even if arbitrary-precision arithmetic is permitted, the function only generates a result when the dimension is or 3 I3 Numerical evaluation and precision tracking When Mathematica performs calculations using arbitrary-precision numbers x, it keeps track of the digits whose correctness it views as guaranteed Precision[x] reports the number of correct base 10 significant digits of x: for instance, if x = d 0 d 1 d d 3 d 4 10 k, the precision is the number of the correct digits in d 0 d 1 d d 3 d 4 Accuracy[x] is the number of correct base 10 digits of x to the right of the decimal point Exact numbers in Mathematica (eg, integers, rational numbers, and algebraic numbers) have Precision equal to To perform certain parts of our analysis (in particular, checking that an eigenvalue of a derivative matrix has negative real part), we need to numerically evaluate exact numbers and expressions We do so using the Mathematica function N N[expr, n] evaluates expr as an arbitrary-precision number at guaranteed precision n When Mathematica performs computations using arbitrary-precision numbers, it maintains precision and accuracy guarantees, the values of which can be accessed using the Precision and Accuracy functions While in principle Mathematica s precision tracking should not make mistakes, there are at least two reasons for exercising caution when using it in proofs First, Mathematica s precision tracking is not based on interval arithmetic, which represents real and complex numbers using exact intervals (in R) and rectangles (in C) that contain the numbers in question, and which relies on theorems that define rules for performing arithmetic and 1 An up-to-date presentation of Gröbner basis algorithms, including many improvements on Buchberger s algorithm, can be found in Cox et al (015) See referencewolframcom/language/tutorial/complexpolynomialsystemshtml for details 3

4 other mathematical operations on these intervals and rectangles that maintain containment guarantees (Alefeld and Herzberger (1983), Tucker (011)) Instead, Mathematica s precision bounds are sometimes obtained using faster methods of the Jenkins-Traub variety (see Section I1), which work correctly in theory but which are difficult to implement perfectly Second, Mathematica s precision tracking is a black box: the specific algorithms it employs are proprietary We contend with these issues by restricting our use of Mathematica s numerical evaluation and precision tracking to a few clearly delineated cases: the evaluation of algebraic numbers, and the basic arithmetic operations of addition, subtraction, multiplication, and division In particular, we do not use Mathematica for precision tracking in the computation of matrix inverses or the solution of linear systems, operations for which interval arithmetic does not generally provide clean answers (Alefeld and Herzberger (1983)) While one could insist that interval arithmetic be used for all non-exact calculations, we chose not to do so II The BEP Centipedenb notebook In this section we describe the main functions from the BEP Centipedenb notebook, which contains all of the procedures we use to analyze BEP dynamics Section II1 describes functions used to prove analytical results, and Section II describes the functions used in numerical analyses and in approximations with error bounds (cf Proposition III4) More details about the use of these functions are provided in the BEP Centipedenb notebook itself Section II3 explains the algorithms used to compute numerical values of rest points of the dynamics and eigenvalues of their derivative matrices Unless stated otherwise, the functions described below take a test-set rule τ {τ all, τ two, τ adj }, a tie-breaking rule β {β min, β stick, β unif } and a length d of the Centipede game as parameters All functions besides the last two are for BEP dynamics with number of trials κ = 1 The BEP Centipedenb notebook includes examples of the use of each of the functions II1 Exact analysis The functions for exact analysis of BEP dynamics in Centipede are as follows: ExactRestPoints the dynamic Uses GroebnerBasis and Reduce to compute the exact rest points of InstabilityOfVertexRestPoint Conducts an analysis of the local stability of the vertex rest point ξ To do this, the function computes the derivative matrix DV (ξ ) of the dynamic and the eigenvalues and eigenvectors of DV(ξ ), where V : aff(ξ) TΞ (see Appendix A) Finally, the function reports whether one can conclude that ξ is unstable The function was not used explicitly in our analysis Instead, we used it to determine the form of the derivative matrix, eigenvalues, and eigenvectors for arbitrary values of d; see Appendix A and Section V below 4

5 LocalStabilityOfInteriorRestPoint Conducts an analysis of the local stability of the interior rest point ξ To do this, the function computes a rational approximation ξ of the exact interior rest point ξ The function then evaluates the eigenvalues of DV(ξ), evaluates the perturbation bound from Proposition III4 (which combines arguments from Appendix B and Section III), and reports whether one can conclude that ξ is asymptotically stable See Section III for further details GlobalStabilityOfInteriorRestPoint Conducts an analysis of the global stability of the interior rest point ξ To do this, the function uses CylindricalDecomposition to determine whether the relevant Lyapunov function (see Sections 5 and 61) is a strict Lyapunov function for the interior rest point ξ on domain Ξ {ξ } We did not use this function in our analysis because it fails to terminate under the settings for exact computation described in Section I3 II Numerical analysis The following functions from the BEP Centipedenb are used for numerical analysis and as subroutines for LocalStabilityOfInteriorRestPoint FloatingPointApproximateRestPoint Computes a floating point approximation of the stable interior rest point of the BEP dynamic See Section II3 for details RationalApproximateRestPoint Computes a rational approximation of the stable interior rest point of the BEP dynamic See Section II3 for details EigenvaluesAtRationalApproximateRestPoint Computes the exact eigenvalues of DV(ξ), where ξ is the rational approximation to the interior rest point obtained from a call to RationalApproximateRestPoint See Section II3 for details NEigenvaluesAtRationalApproximateRestPoint Computes the eigenvalues of DV( ξ) using arbitrary-precision arithmetic, where ξ is a 16-digit precision approximation to the rational point computed using RationalApproximateRestPoint See Section II3 for details NumericalGlobalStabilityOfInteriorRestPointLyapunov Evaluates the time derivative Λ(ξ) = Λ(ξ) V(ξ) at a floating-point approximation Λ of the appropriate candidate Lyapunov function L for the interior rest point ξ, reporting instances in which the time derivative is not negative should any exist The (presumably large number of) states ξ at which to evaluate Λ(ξ) is chosen by the user NumericalGlobalStabilityOfInteriorRestPointNDSolve Computes numerical solutions to the BEP dynamic from initial conditions provided by the user, and reports whether any of these numerical solutions fails to converge to a neighborhood of the interior rest point ξ 5

6 NDSolveMeanDynamics Uses Mathematica s NDSolve function to compute a numerical solution to the BEP dynamic from an initial condition provided by the user The solution is computed until the time at which the norm of the law of motion is sufficiently small, where what constitutes sufficiently small can be chosen by the user The function also graphs the components of the state as a function of time, and reports the terminal point and the time at which this point is reached FloatingPointApproximateRestPointTestAllMinIfTieWithBIAgents Computes a floating point approximation of the stable interior rest point of the dynamics in a population consisting of mass b of backward induction agents and mass (1 b) of BEP(τ all, 1, β min ) agents The value of b is specified by the user This function was used to produce Figures 5 and 6 See Section II3 for details FloatingPointApproximateRestPointTestTwoMinIfTieWithBIAgents Computes a floating point approximation of the stable interior rest point of the dynamics in a population consisting of mass b of backward induction agents and mass (1 b) of BEP(τ two, 1, β min ) agents The value of b is specified by the user This function was used to produce Figure 7 See Section II3 for details FloatingPointApproximateRestPointTestAllMinIfTieManyTrials Uses Mathematica s FindRoot function to compute a floating point approximation of the stable interior rest point of the BEP(τ all, κ, β min ) dynamic, where the number of trials κ is specified by the user This function was used in producing Figures 8 and 9 NDSolveMeanDynamicsTestAllMinIfTieManyTrials Uses Mathematica s NDSolve function to compute a numerical solution of the BEP(τ all, κ, β min ) dynamic, where the number of trials κ and the initial condition of the solution are specified by the user The solution is computed until the time at which the norm of the law of motion is sufficiently small, where what constitutes sufficiently small can be chosen by the user The function also graphs the components of the state as a function of time, and reports the terminal point and the time at which this point is reached The function was used in producing Figures 8, 9, and 10 II3 More on computation of approximate rest points and eigenvalues The BEP Centipedenb notebook computes approximate rest points of BEP(τ, 1, β) dynamics using the Euler method: {ξ t } T t=0 is computed starting from an initial condition ξ 0 by iteratively applying (1) ξ t+1 = ξ t + h V (ξ t ), where V : R s R s is the (extended) law of motion of the dynamics and h is the step size of the algorithm This algorithm is run in two sequential stages, to be described next When one of the first three FloatingPointApproximateRestPoint functions from Section II is called, algorithm (1) is run using IEEE 754 Standard double-precision 6

7 floating-point arithmetic The step size of the algorithm is set to h = 4, and the initial condition is ξ 0 = (x 0, y 0 ) Ξ = (X, Y), where x 0 and y 0 are the barycenters of simplices X and Y Several thousand iterations of (1) are run, and the output of each iteration is projected onto Ξ to minimize the accumulation of roundoff errors from the floating-point calculation The floating-point numbers obtained in this way are very close to the exact quantities they approximate, but their digits (ie, the values of the d i in x = d 0 d 1 d d 3 d 4 10 k ) may all be wrong, especially in small numbers, since many of the exact numbers we aim to approximate lie outside the range of IEEE 754 double-precision 3 To address this issue, the function RationalApproximateRestPoint begins with a call to FloatingPointApproximateRestPoint, and then uses the output of this procedure to create the initial condition for a second stage that employs rational arithmetic This initial condition is the rational point in Ξ that lies closest to the floating-point output of the first stage The step size h is set to 1 in the second stage, since overshooting is no longer a problem in the neighborhood of the exact rest point Increment (1) is executed repeatedly using rational arithmetic until it locates a rational point ξ T that is an approximate fixed point of (1), in the sense that ξ T and ξ T+1 = ξ T + V (ξ T ) agree with 6 digits of precision for numbers greater or equal to 10 4, or 3 digits of precision for smaller numbers This agrees with the format we use to report rest points in the tables in Section VIII NEigenvaluesAtRationalApproximateRestPoint computes the eigenvalues of DV( ξ) using arbitrary-precision arithmetic, where ξ is a 16-digit precision approximation to the rational point computed by calling RationalApproximateRestPoint The use of arbitrary precision allows us to keep track of the precision of the computed eigenvalues Proposition III4 provides a bound on the distances between the eigenvalues of DV(ξ) and the eigenvalues of DV(ξ ) In the tables in Section IX, the reported eigenvalues, which are arbitrary-precision approximations to the (algebraic-valued) eigenvalues of DV(ξ), are shown with 5 digits of precision for numbers greater or equal to 1, 4 digits of precision for numbers greater or equal to 10, and 3 digits of precision for smaller numbers III Dimension reduction for local stability analysis This section presents the dimension reduction step used to reduce the computational demands of computing eigenvalue perturbation bounds, and presents a version of this bound (Proposition III4) that incorporates all of the simplifications introduced here and in Appendix B Write a = s 1, b = s, and s = s 1 + s, and recall that d = s The computations we use to prove local stability of the interior rest point require calculations involving the derivative matrices DV (ξ) R s s that quickly become very computationally demanding as the size of the matrix grows Since we are only interested in the action of DV (ξ) on the d-dimensional subspace TΞ, it should be possible to perform the desired calculations 3 For example, note that the IEEE 754 double-precision representation of numbers such as and (both of which appear in Table 1 below) is 0, since both numbers are well below , which is the smallest positive IEEE 754 double-precision number 7

8 using matrices in R d d We now show explicitly how this is done The analysis is a simple extension of arguments from Sandholm (007, p 661) Define the orthonormal matrix R R s s by R = a a a(a 1) + 1 a a a(a 1) a a a(a 1) a a a a a(a 1) a a a a(a 1) a(a 1) a a + 1 a(a 1) a a a a a(a 1) a(a 1) a a a(a 1) + 1 a 0 0 a a a a 0 0 a a 0 0 b b + 1 b b b b b(b 1) b(b 1) b(b 1) b b b b + 1 b(b 1) b(b 1) b b b b b(b 1) b(b 1) b b b b b b + 1 b(b 1) b(b 1) b(b 1) 0 0 b b b b b b b b Define the matrix J R d s J = Define R R d s by R = JR In words, R is R with the last row in each block removed Let e a denote the last standard basis vector in R a The upper diagonal block of R rotates { span{1, e a } = span 1 a 1, a a 1 ( ea 1 a 1)} R a about its orthogonal complement by an angle of cos 1 ( 1 a ) It is easy to verify that this block maps 1 to ae a, and so, by virtue of being orthonormal, maps TΞ 1 isometrically to R a = {x Ra : x a = 0} Likewise, the lower diagonal block of R maps 1 to be b and maps TΞ isometrically to R b = {y Rb : y b = 0} Altogether, premultiplying z R s by R double-rotates z so that its TΞ component lies in R a Rb Then premultiplying the result by J removes the now-superfluous final coordinates of each block Recall from Appendix A that the vector field V maps aff(ξ) to TΞ, so that DV(ξ)z TΞ 8

9 for all ξ Ξ and z TΞ, and that the extension V of V maps R s to itself, so that DV (ξ) also maps R s to itself Proposition III1 If ξ aff(ξ), then DV(ξ) and R DV (ξ) R have the same eigenvalues, including multiplicities Proposition III1 is an immediate corollary of the following lemma: Lemma III Suppose that M R s s maps TΞ into itself, and let z C s be an element of the complexification of TΞ Then (λ, z) is an eigenvalue/eigenvector pair of M if and only if (λ, Rz) is an eigenvalue/eigenvector pair for RM R Proof The proof follows Sandholm (007) To start, recall from Appendix A that Φ R s s is the orthogonal projection of R s onto TΞ, and note the following geometrically obvious facts, each of which can be verified by direct computation: () (3) R R = R J JR = Φ R s s, R R = JRR J = JJ = I R d d If Mz = λz, then since Mz and z are in the complexification of TΞ, we have RMΦz = λ RΦz; thus () and (3) imply that RM R Rz = λ R R Rz = λ Rz Conversely, if RM R Rz = λ Rz, then () implies that RMΦz = RMz = λ Rz, and so that Mz = λz The following result is also needed to obtain the eigenvalue bound Proposition III3 For M, M R s s, RM R RM R 4 M M Proof Using the submultiplicativity of matrix norms and the orthonormality of R, RM R RM R RMR RM R = R(M M )R R R R R(M M )R R 4 M M Using the results above and the arguments from Appendix B, including the definition = max i S max k S j S we obtain the following result: V i ξ j ξ k (1,, 1 1,, 1), Proposition III4 Suppose that RDV (ξ) R is complex diagonalizable with RDV (ξ) R = Q diag(λ) Q 1, and let λ be an eigenvalue of DV(ξ ) Then there is an eigenvalue λ i of DV(ξ) such that (4) λ λ i < 8 d d/ 1 tr( Q Q) d/ det( Q) ξ k ξ k k S 9

10 When the function InstabilityOfVertexRestPoint from the BEP Centipedenb notebook is called, the eigenvectors in the matrix Q are chosen to have the Euclidean norm 1, as this tends to lower the bound on the condition number of Q (see Guggenheimer et al (1995)) If this normalization were performed exactly, then we would have tr( Q Q) = d, allowing us to simplify inequality (4) However, because InstabilityOfVertexRestPoint performs the normalization after converting the entries of Q to arbitrary precision numbers, it uses the original inequality (4) Of course, the effect of this choice on the bound we obtain is essentially nil IV Time until convergence under different test-set rules Figures 1,, and 3 present numerical solutions to the BEP(τ, 1, β min ) dynamics with τ = τ all, τ two, and τ adj in a Centipede game of length d = 10 All solutions have initial condition ξ = (x, y) = ((99, 01, 0, ), (99, 01, 0, )) 4 The computation of the solution is cut off when it enters a Euclidean ball of radius 10 3 centered at the stable rest point ξ of each dynamic The numbers of time units required until the ball is reached are 1011 under τ all, 3981 under τ two, and 1507 under τ adj, as suggested in Section 61 Figures 4, 5, and 6 present solutions as above, but for a Centipede game of length d = 0 In this case the numbers of time units required until the ball of radius 10 3 around ξ is reached are 98 under τ all, 7640 under τ two, and 4165 under τ adj, again agreeing with the claim in Section 61 It is noteworthy that the time until convergence under BEP(τ all, 1, β min ) is faster in the game of length 0 than in the game of length 10 Judging from Figures 1 and 4, it appears that when revising agents test all strategies, having more strategies to test makes them abandon strategy 1 more quickly in favor of more cooperative strategies, which in turn increases the chances that still more cooperative strategies will be chosen during subsequent revisions 4 The results are similar for other choices of ξ with x 1 = y 1 = 99 10

11 10 x 1 x 4 10 y 1 y 4 08 x x 3 x 5 x 6 08 y y 3 y 5 y (i) population (ii) population Figure 1: Solution to the BEP(τ all, 1, β min ) dynamic from initial condition ξ in Centipede of length d = x 1 x 4 10 y 1 y 4 08 x x 3 x 5 x 6 08 y y 3 y 5 y (i) population (ii) population Figure : Solution to the BEP(τ two, 1, β min ) dynamic from initial condition ξ in Centipede of length d = x 1 x 4 10 y 1 y 4 08 x x 3 x 5 x 6 08 y y 3 y 5 y (i) population 1 (ii) population Figure 3: Solution to the BEP(τ adj, 1, β min ) dynamic from initial condition ξ in Centipede of length d = 10 11

12 10 10 x 1 x 5 x 9 y 1 y 5 y 9 08 x x 3 x 6 x 7 x 10 x y y 3 y 6 y 7 y 10 y 11 x 4 x 8 y 4 y (i) population (ii) population Figure 4: Solution to the BEP(τ all, 1, β min ) dynamic from initial condition ξ in Centipede of length d = x 1 x 5 x 9 y 1 y 5 y 9 08 x x 3 x 6 x 7 x 10 x y y 3 y 6 y 7 y 10 y 11 x 4 x 8 y 4 y (i) population (ii) population Figure 5: Solution to the BEP(τ two, 1, β min ) dynamic from initial condition ξ in Centipede of length d = x 1 x 5 x 9 y 1 y 5 y 9 08 x x 3 x 6 x 7 x 10 x y y 3 y 6 y 7 y 10 y 11 x 4 x 8 y 4 y (i) population (ii) population Figure 6: Solution to the BEP(τ adj, 1, β min ) dynamic from initial condition ξ in Centipede of length d = 0 1

13 V Analyses of repulsion from the backward induction state V1 Test-two, stick-if-tie: Under the BEP(τ two, 1, β stick ) dynamic, DV (ξ ) = m m m m m m m m m n n n n For d 3, the eigenvalues of DV(ξ ) with respect to TΞ and the bases for their eigenspaces are: (5) (6) (7) (8) 0, { ε ε j : j {3,, s } } if d 4; 1, { m δ δ i : i {3,, s 1 } } ; { λ ( 1 m m ), ( λ, λ,, λ } m m 1, 1,, 1 n n ) ; and { λ ( 1 m m ), ( λ +, λ +,, λ } + m m 1, 1,, 1 n n ) The eigenvectors in (5) span the center subspace E c of the linear equation ż = DV(ξ ) z, while the eigenvectors in (6) and (7) span the stable subspace E s The normal vector to the hyperplane E c E s is the orthogonal projection onto TΞ of the auxiliary vector which satisfies z aux = 1 λ δ 1 ε 1 (z ) (δ i δ 1 ) = 1 λ > 0 for i S 1 {1}, and (z ) (ε j ε 1 ) = 1 > 0 for j S {1} Since the remaining eigenvalue, from (8), is positive, the arguments used in the appendix for BEP(τ all, 1, β stick ) imply that ξ is a repellor 13

14 V Test-adjacent, stick-if-tie: Under the BEP(τ adj, 1, β stick ) dynamic, DV(x, y ) = For d 3, the eigenvalues of DV(ξ ) with respect to TΞ and the bases for their eigenspaces are: (9) (10) (11) 0, λ λ , , { (δ δ 3 ) ε 1 + ε } { ε ε j : j {3,, s } } if d 4 { δ 3 δ i : i {4,, s 1 } } if d 5; { } ( λ, λ, 0,, 0 1, 1, 0, 0) ; and { } ( λ +, λ +, 0,, 0 1, 1, 0, 0) The eigenvectors in (9) span the center subspace E c of the linear equation ż = DV(ξ ) z, while the eigenvector in (10) spans the stable subspace E s The normal vector to the hyperplane E c E s is the orthogonal projection onto TΞ of the auxiliary vector which satisfies z aux = ( 1 λ 1 ) δ1 1 δ ε 1 (z ) (δ δ 1 ) = (z aux) (δ δ 1 ) = 1 λ > 0, (z ) (δ i δ 1 ) = (z aux) (δ i δ 1 ) = λ > 0 for i S 1 {1, }, and (z ) (ε j ε 1 ) = (z aux) (ε j δ 1 ) = 1 > 0 for j S {1} Since the remaining eigenvalue, from (11), is positive, the arguments used in the appendix for BEP(τ all, 1, β stick ) imply that ξ is a repellor 14

15 VI Formulas for BEP(τ, 1, β) dynamics in Centipede This section provides explicit formulas for BEP(τ, 1, β) dynamics in the Centipede game for the cases considered in the paper that is, for τ {τ all, τ two, τ adj } and β {β min, β stick, β unif } These are the formulas implemented in the BEP Centipedenb notebook VI1 Test-all BEP(τ all, 1, β min ): ẋ i = s k=i ẏ j = s 1 s 1 k=j+1 BEP(τ all, 1, β stick ): ẋ i = + s k=i s 1 q=i+1 i 1 + x i ẏ j = + s 1 k=j+1 s q=j+1 y k i m=1 y m s1 i + i 1 k 1 y k y l i k k m=1 y m s1 i x i, x k (x 1 + x ) s 1 + (x 1 ) s y 1 if j = 1, j+1 x k y k i m=1 i 1 x q m=1 k 1 y k m=1 x m s j + y m s1 i + i k 1 y k j k 1 j k+1 k x k x l x m s j y j i 1 y l i k k j+1 x k x m s j + y q q=1 i 1 x q k 1 y k i k k y l y m s1 i 1 + m=1 m=1 y m s1 k 1 x i, j 1 y q q=1 m=1 i k k y l y m s1 i + m=1 j k 1 j k+1 k x k x l x m s j + j k 1 j k+ k x k x l x m s j y j (x 1) s + m=1 j k 1 k x k x l x m s k y j m=1 m=1 otherwise 15

16 BEP(τ all, 1, β unif ): ẋ i = ẏ j = s j=i s 1 h=j+1 VI Test-two BEP(τ two, 1, β min ): ẋ i = 1 s s 1 1 ẏ j = 1 s s 1 BEP(τ two, 1, β stick ): ẋ i = 1 s s 1 1 ẏ j = 1 s s 1 y j i j=1 y j s1 i + j+1 x h x h s j i 1 h=1 s 1 h=i+1 j 1 h=1 s h=j+1 i 1 h=1 s 1 h=i+1 j 1 h=1 s h=j+1 h=1 s k=h+1 s i 1 + xs 1 y k + k=i s 1 k=h+ s 1 k 1 y k s + j=1 y j s1 k 1 j=0 h=1 ( ) s 1 j k 1 y k 1 k j j + 1 y l s1 k 1 j x i, h=0 j k 1 x k x h s k ( ) s k x h k 1 k h h + 1 x l s k h y j h k 1 y k y l (x i + x h ) + i i 1 k 1 y k y l + y k y l + i 1 h+1 k 1 x k + x k x l (y j + y h ) + j+1 k=j+1 s k=h+1 s y k + k=i s 1 k=h+ s 1 x k x l + h k 1 j k 1 x k x l + j y k x k h 1 y k y l (x i + x h ) + x i (x i + x h ) x i, (y j + y h ) y j y k + i i 1 k 1 i 1 y k y l + y k y l (x i + x h ) + x i h+1 k 1 x k + x k x l (y j + y h ) + y j j+1 k=j+1 x k x l + h x k j k 1 x k x l (y j + y h ) + y j + j y k x i, x k y j 16

17 BEP(τ two, 1, β unif ): ẋ i = 1 s s 1 1 ẏ j = 1 s s 1 i 1 h=1 s 1 h=i+1 j 1 h=1 s h=j+1 VI3 Test-adjacent s k=h+1 s k=i s 1 k=h+ s 1 h k 1 y k + y k y l + 1 h 1 y k (x i + x h ) + i i 1 k 1 y k y l + y k y l + 1 i 1 y k (x i + x h ) x i, h+1 k 1 x k + x k x l + 1 h x k (y j + y h ) + j+1 j k 1 x k x l + x k x l + 1 j x k (y j + y h ) y j k=j+1 Let c p i equal if i {1, s p } and equal 1 otherwise, ie, c p i = 1 + 1[i {1, s p }] BEP(τ adj, 1, β min ): ẋ i = ẏ j = 1 1[i = 1] + 1 1[i = s1 ] 1 1[j = 1] + 1 1[j = s ] i 1 h=i 1 i+1 h=i+1 j 1 h=j 1 j+1 h=j+1 s k=h+1 s y k + k=i s 1 k=h+ h k 1 y k y l ( c 1 i x i + c 1 h x h) + i i 1 k 1 y k y l + y k y l + h+1 k 1 x k + s 1 j+1 k=j+1 x k x l + i 1 y k x k x l ( c j y j + c h y h) + j k 1 x k x l + j x k ( c 1 i x i + c 1 h x h) x i, ( c j y j + c h y h) y j 17

18 BEP(τ adj, 1, β stick ): ẋ i = ẏ j = 1 1[i = 1] + 1 1[i = s1 ] BEP(τ adj, 1, β unif ): ẋ i = ẏ j = 1 1[j = 1] + 1 1[j = s ] 1 1[i = 1] + 1 1[i = s1 ] 1 1[j = 1] + 1 1[j = s ] i 1 h=i 1 i+1 h=i+1 j 1 h=j 1 j+1 h=j+1 i 1 s k=h+1 s h=i 1 i+1 h=i+1 j 1 h=j 1 j+1 h=j+1 y k + k=i s 1 k=h+ k=j+1 h k 1 i i 1 k 1 y k y l + h+1 k 1 x k + s 1 j+1 s k=h+1 s y k + k=i s 1 k=h+ x k x l + ( y k y l c 1 i x i + c 1 h x ) h 1 h + c 1 i x i y k ( y k y l c 1 i x i + c 1 h x ) h + c 1 i x i ( x k x l c j y j + c h y ) h + c j y j h + i 1 x k j k 1 ( x k x l c j y j + c h y ) h + c j y j h k 1 y k y l + 1 h 1 y k i i 1 k 1 y k y l + y k y l + 1 h+1 k 1 x k + x k x l + 1 h x k s 1 j+1 k=j+1 x k x l + j k 1 x k x l + 1 i 1 ( c 1 i x i + c 1 h x h) + y k + y k j x i, x k y j ( c 1 i x i + c 1 h x h) x i, ( c j y j + c h y h) + j x k ( c j y j + c h y h) y j VII Multinomial formulas for BEP(τ, κ, β) dynamics The general formula (B) for the BEP(τ, κ, β) dynamic explicitly lists each of the κ strategies played an agent s opponents when an agent tests a strategy i in his test set We can obtain a formula with far fewer terms by instead working with the distribution of opponents strategies when the agent tests strategy i Using such formulas is essential for numerical computations when κ is not small To express (B) in this form we introduce a number of definitions Let Z sq,κ + = z Zsq + : z j = κ j S q 18

19 denote the set of possible (unnormalized) empirical distributions of opponents strategies when a population p agent tests one of his own strategies κ times When the state of population q is ξ q Ξ q, the probability that empirical distribution z occurs is the multinomial probability ( ) M p,κ (z, ξ q κ ) = (ξ q 1 z 1 z )z1 (ξ q ) z s q s q s q And if a population p agent faces empirical distribution z when testing strategy i S p, his total payoff is π p i (z) = j S q U p ij z j Therefore, if we let Π p,κ (ξ q ) be a random variable representing the total payoff obtained if i strategy i S p is tested κ times when the state of the opposing population is ξ q, then the distribution of Π p,κ (ξ q ) is i P ( ) Π p,κ (ξ q ) = w p i i = M p,κ (z, ξ q ) z Z sq,κ + : πp i (z)=w p i We use the notation above to obtain our new expression for BEP dynamics Let W p,κ i = {π p i (z): z Zsq,κ + } denote the set of possible test results for strategy i S p in κ trials Also, for R p S p, write W p,κ = R p k R p Wp,κ Then we can express the BEP(τ, κ, β) dynamic as k ( ) (1) ξ p = ξ i j τ p j j S (Rp p,κ ) p R p S Π (ξ q ) = w p k k β p (w p, R p ) ji ξp i p w p W p,κ R p k R p P The general formula (1) becomes simpler if particular test-set and tie-breaking rules are chosen For instance, under the convention that an empty product evaluates to 1, the BEP(τ all, κ, β min ) dynamic can be expressed as ẋ i = ẏ j = w 1 i W1,κ i w j W,κ j P( Π 1,κ (y) = w 1 i i P( Π,κ (x) = w i j ) i 1 ) j 1 P ( ) s1 Π 1,κ (y) < w 1 k i P ( ) Π 1,κ (y) w 1 l i x i, l=i+1 P ( ) s Π,κ (x) < w k i P ( ) Π,κ (x) w l i y j l=j+1 19

20 VIII Approximate components of interior rest points The tables in this section present approximate components of the unique interior rest points of BEP(τ, 1, β) dynamics in Centipede games of lengths up to d = 0 Dashed lines in the tables separate cases in which the values were originally computed exactly from those in which the values were computed numerically Tables 1,, and 3 present the approximate rest points of the BEP(τ all, 1, β) dynamics with β = β min, β stick, and β unif The approximate rest points of the BEP(τ two, 1, β) dynamics are presented in Tables 4 6, and those of the BEP(τ adj, 1, β) dynamics are presented in Tables 7 9 The main text contains graphs of the rest points as a function of d for the three cases with tie-breaking rule β min Graphs for the remaining six cases appear here as Figures 7 1 0

21 VIII1 Test-all p [6] [5] [4] [3] [] [1] [0] q [6] [5] [4] [3] [] [1] [0] Table 1: The interior rest point of the BEP(τ all, 1, β min ) dynamic for Centipede of lengths d {3,, 0} p denotes the penultimate player, q the last player The dashed lines separated exact (d 6) from numerical (d 7) results 1

22 p [6] [5] [4] [3] [] [1] [0] q [6] [5] [4] [3] [] [1] [0] Table : The interior rest point of the BEP(τ all, 1, β stick ) dynamic for Centipede of lengths d {,, 0} p denotes the penultimate player, q the last player The dashed lines separated exact (d 5) from numerical (d 6) results

23 p [6] [5] [4] [3] [] [1] [0] q [6] [5] [4] [3] [] [1] [0] Table 3: The interior rest point of the BEP(τ all, 1, β unif ) dynamic for Centipede of lengths d {,, 0} p denotes the penultimate player, q the last player The dashed lines separated exact (d 6) from numerical (d 7) results 3

24 d (i) penultimate mover d (ii) last mover Figure 7: The interior rest point of Centipede under the BEP(τ all, 1, β min ) dynamic for game lengths d = 3,, 0 Markers,,, and, represent weights on strategies [0], [1], [], and [3] (continue at all decision nodes; stop at the last, second-to-last, or third-to-last decision node) Other weights are less than 10 8 The dashed line separates exact (d 6) and numerical (d 7) results d (i) penultimate mover d (ii) last mover Figure 8: The interior rest point of Centipede under the BEP(τ all, 1, β min ) dynamic for game lengths d = 3,, 0 Markers,,, and, represent weights on strategies [0], [1], [], and [3] (continue at all decision nodes; stop at the last, second-to-last, or third-to-last decision node) Other weights are less than 10 8 The dashed line separates exact (d 6) and numerical (d 7) results 4

25 VIII Test-two p [7] [6] [5] [4] [3] [] [1] [0] q [7] [6] [5] [4] [3] [] [1] [0] Table 4: The interior rest point of the BEP(τ two, 1, β min ) dynamic for Centipede of lengths d {3,, 0} p denotes the penultimate player, q the last player The dashed lines separated exact (d 8) from numerical (d 9) results 5

26 p [7] [6] [5] [4] [3] [] [1] [0] q [7] [6] [5] [4] [3] [] [1] [0] Table 5: The interior rest point of the BEP(τ two, 1, β stick ) dynamic for Centipede of lengths d {,, 0} p denotes the penultimate player, q the last player The dashed lines separated exact (d 8) from numerical (d 9) results 6

27 p [7] [6] [5] [4] [3] [] [1] [0] q [7] [6] [5] [4] [3] [] [1] [0] Table 6: The interior rest point of the BEP(τ two, 1, β unif ) dynamic for Centipede of lengths d {,, 0} p denotes the penultimate player, q the last player The dashed lines separated exact (d 8) from numerical (d 9) results 7

28 d (i) penultimate mover d (ii) last mover Figure 9: The stable rest point of Centipede under the BEP(τ two, 1, β min ) dynamic for game lengths d = 3,, 0 Markers,,,,, and represent weights on strategies [0], [1], [], [3], [4], and [5] Other weights are less than 10 4 The dashed line separates exact (d 8) and numerical (d 9) results d (i) penultimate mover d (ii) last mover Figure 10: The stable rest point of Centipede under the BEP(τ two, 1, β min ) dynamic for game lengths d = 3,, 0 Markers,,,,, and represent weights on strategies [0], [1], [], [3], [4], and [5] Other weights are less than 10 4 The dashed line separates exact (d 8) and numerical (d 9) results 8

29 VIII3 Test-adjacent p [7] [6] [5] [4] [3] [] [1] [0] q [7] [6] [5] [4] [3] [] [1] [0] Table 7: The interior rest point of the BEP(τ adj, 1, β min ) dynamic for Centipede of lengths d {3,, 0} p denotes the penultimate player, q the last player The dashed lines separated exact (d 7) from numerical (d 8) results 9

30 p [7] [6] [5] [4] [3] [] [1] [0] q [7] [6] [5] [4] [3] [] [1] [0] Table 8: The interior rest point of the BEP(τ adj, 1, β stick ) dynamic for Centipede of lengths d {,, 0} p denotes the penultimate player, q the last player The dashed lines separated exact (d 6) from numerical (d 7) results 30

31 p [7] [6] [5] [4] [3] [] [1] [0] q [7] [6] [5] [4] [3] [] [1] [0] Table 9: The interior rest point of the BEP(τ adj, 1, β unif ) dynamic for Centipede of lengths d {,, 0} p denotes the penultimate player, q the last player The dashed lines separated exact (d 7) from numerical (d 8) results 31

32 d (i) penultimate mover d (ii) last mover Figure 11: The stable rest point of Centipede under the BEP(τ adj, 1, β min ) dynamic for game lengths d = 3,, 0 Markers,,,,, and represent weights on strategies [0], [1], [], [3], [4], and [5] Other weights are less than 10 4 The dashed line separates exact (d 6) and numerical (d 7) results d (i) penultimate mover d (ii) last mover Figure 1: The stable rest point of Centipede under the BEP(τ adj, 1, β min ) dynamic for game lengths d = 3,, 0 Markers,,,,, and represent weights on strategies [0], [1], [], [3], [4], and [5] Other weights are less than 10 4 The dashed line separates exact (d 7) and numerical (d 8) results 3

33 IX Approximate eigenvalues of DV(ξ ) The tables in this section show approximate eigenvalues of the derivative matrix DV(ξ ) at the interior rest point ξ of BEP(τ, 1, β) dynamics in Centipede games of lengths up to d = 0 Tables 10, 11, and 1 present the approximate eigenvalues of DV(ξ ) for the BEP(τ all, 1, β) dynamics with β = β min, β stick, and β unif The approximate eigenvalues for BEP(τ two, 1, β) dynamics are in Tables 13 15, and those for BEP(τ adj, 1, β) dynamics are in Tables IX1 Test-all d = 3 1 ± d = ± 377 i 8589 ± 377 i d = ± 384 i 8645 ± 384 i 1 d = ± 384 i 8645 ± 384 i 1 ± i d = ± 384 i 8645 ± 384 i 1 ± i 1 d = ± 384 i 8645 ± 384 i 1 ± i 1 1 d = ± 384 i 8645 ± 384 i 1 ± i d = ± 384 i 8645 ± 384 i 1 ± i d = ± 384 i 8645 ± 384 i 1 ± i Table 10: Approximate eigenvalues of DV(ξ ) for the BEP(τ all, 1, β min ) dynamic The symbol 1 is used as a shorthand for d = 8090 ± 4468 i d = ± 5843 i 1 d = ± 399 i 7756 ± 3807 i d = ± 393 i 773 ± 3746 i 9989 d = ± 393 i 773 ± 3746 i d = ± 393 i 773 ± 3746 i d = ± 393 i 773 ± 3746 i d = ± 393 i 773 ± 3746 i d = ± 393 i 773 ± 3746 i Table 11: Approximate eigenvalues of DV(ξ ) for the BEP(τ all, 1, β stick ) dynamic The symbol 1 is used as a shorthand for