Rock-Paper-Scissors Game
4 Comparisons
1 Problem
▶ Rock-Paper-Scissors(RPS) game: a metaphor of
3-species competition to show cyclic dominance e.g.
in ecology or evolutionary biology.
▶ Proposal: investigate stochastic and deterministic
methods for the model to understand why
sometimes they behave similarly, but sometimes
differently.
▶ Summary: the three species dominate in turns,
their evolution forms periodic orbits. These orbits
are our focus.
▶ Reason for difference: Assumptions of ODE modelling infinite and continuous system, while simulations - finite
and discrete population.
▶ Observations:
Fig.1 - Numerical solutions of ODEs forms an attracting
β
periodic orbit, i.e. a limit cycle when 0 < µ < 18 . In
stochastic simulations, cycles are also visible.
Fig.2 - Compare period of limit cycle to period of
simulated quasi-periodic cycle. N, µ are variables, while
β = 0.5 is fixed.
0.9
2 Model
ODEs result
SSA result of N=28
SSA result of N=216
0.8
0.7
Fig.1
solutions,
0.6
0.5
2
(1)
0.4
y : x beats y
y : x mutates into y
0.3
0.2
A
B
C 

A 
0
−1 − β
1



P =



B 
1
0
−1 − β 




C −1 − β
1
0
B(Paper)

0.1
0
0
0.2
0.4
0.6
0.8
1
y
1
A(Rock)
C(Scissors)
▶ Parameter β : unbalanced payoff, matrix P is
payoff matrix of the game.
▶ Symmetry: The model is rotationally symmetric.
5
1
A+B −
→
B + B,
1
B+C −
→
C + C,
A+B+B −
→ A + A + B,
1
1
C +A−
→
A + A,
1+β
A + B + B −−→ B + B + B,
1+β
1
A+A+C −
→ A + C + C,
A + A + C −−→ A + A + A,
B+C +C −
→ B + B + C,
1+β
1
B + C + C −−→ C + C + C,
References
[1] Mobilia, Mauro. Oscillatory dynamics in rock–paper–scissors
games with mutations. Journal of Theoretical Biology 264.1
(2010): 1-10.
[2] Gillespie, Daniel T. Exact Stochastic Simulation of Coupled
Chemical Reactions. The Journal of Physical Chemistry 81.25
(1977): 2340– 2361
Supervisors: Jonathan Dawes
Tstoch
TODE
4
10
Period:T
▶ Deterministic method - ODEs[1]
a(t), b(t), c(t) are the densities of the three species
(A, B , C ), evolving in time according to:
ȧ = a[c − (1 + β)b + β(ab + bc + ac)] + µ(1 − 3a),
ḃ = b[a − (1 + β)c + β(ab + bc + ac)] + µ(1 − 3b),
ċ = c[b − (1 + β)a + β(ab + bc + ac)] + µ(1 − 3c).
with a(t) + b(t) + c(t) = 1.
▶ Stochastic method - simulations[2] of the
chemical reactions (1) and the following:
Period T against varing µ when N is large (N=217)
10
3 Methods
Qian Yang
ODEs
β
β
0 < µ = 108
< 18
.
Stochastic simulations, N = 28, 216,
β
.
µ = 108
y
▶ Parameter µ: mutation rate, eq (1) are
corresponding chemical reactions.
µ
µ
µ
A−
→ B, B −
→ C, C −
→ A,
µ
µ
µ
A−
→ C, B −
→ A, C −
→ B.
x
x
Unbalanced Rockpaperscissor with mutation
3
10
2
10
1
10
Region I
Region II
Region III
Fig.2 Difference
between solutions
of ODEs and
stochastic simulations in three
regions
0
10 10
10
8
10
6
10
4
µ
10
2
10
0
10
5 Discussion and Conclusion
▶ Research Object: Period of the cycles TODE and TStoch
shown in Fig.2.
1
▶ Region I: µc > µ ≫ N log
N , TODE ≈ TStoch ∝ − log µ,
deterministic dynamic wins. It can be proved theoretically
by composing local-global map.
1
1
▶ Region II: 0 < µ ≪ N log N , TStoch ∝ N µ , Markov process
determines dynamics. Its proof follows analysis of a 2-D
birth-death process.
1
c0
▶ Region III: µ ∼ N log N , TStoch ∝ − log µ + N µ , two
dynamics run a tight race. Asymptotic phase and
Stochastic Differential Equations (SDEs) are applied to
prove the conclusion.
Tim Rogers
{Q.Yang2,J.H.P.Dawes,T.C.Rogers}@bath.ac.uk
Department of Mathematical Science, University of Bath