```
# Gangsta's paradise game from the paper
library(RelationalContracts)
g = rel_game() %>%
# PD state
rel_state("x0", A1=list(a1=c("C","D")), A2=list(a2=c("C","D")),
pi1=case_when(
a1 == "C" & a2== "C" ~ 2,
a1 == "D" & a2== "C" ~ 3,
a1 == "C" & a2== "D" ~ -10,
a1 == "D" & a2== "D" ~ 0
),
pi2=case_when(
a1 == "C" & a2== "C" ~ 2,
a1 == "D" & a2== "C" ~ -10,
a1 == "C" & a2== "D" ~ 3,
a1 == "D" & a2== "D" ~ 0
)
) %>%
rel_state("x.paradise", pi1=2, pi2=2) %>%
rel_transition("x0","x.paradise",a1="C",a2="C") %>%
rel_compile()
# Solve a T-RNE
g = g %>% rel_T_rne(T=99, delta=0.5, rho=1, save.history = TRUE, tie.breaking = "slack")
get_T_rne_history(g) %>%
filter(x!="x.paradise", t <= 8) %>%
arrange(t) %>%
select(t,ae.lab,r1,r2,U,v1,v2)
# Check if we can have an RNE if we split the state
# We now split x0 into two states x0 and x1
# As long as paradise is not yet reached, we alternate
# each period between x0 and x1
g = rel_game() %>%
# PD state
rel_states(c("x0","x1"), A1=list(a1=c("C","D")), A2=list(a2=c("C","D")),
pi1=case_when(
a1 == "C" & a2== "C" ~ 2,
a1 == "D" & a2== "C" ~ 3,
a1 == "C" & a2== "D" ~ -10,
a1 == "D" & a2== "D" ~ 0
),
pi2=case_when(
a1 == "C" & a2== "C" ~ 2,
a1 == "D" & a2== "C" ~ -10,
a1 == "C" & a2== "D" ~ 3,
a1 == "D" & a2== "D" ~ 0
)
) %>%
rel_state("x.paradise", pi1=2, pi2=2) %>%
rel_transition("x0","x1",a1="D") %>%
rel_transition("x0","x1",a2="D") %>%
rel_transition("x1","x0",a1="D") %>%
rel_transition("x1","x0",a2="D") %>%
rel_transition("x0","x.paradise",a1="C",a2="C") %>%
rel_compile()
g = g %>% rel_T_rne(T=99, delta=0.5, rho=1, save.history = TRUE, tie.breaking = "slack")
get_T_rne_history(g) %>%
filter((x=="x0" & t %% 2 == 1) | (x=="x1" & t %% 2 == 0), t <= 10) %>%
arrange(t) %>%
select(t,x,ae.lab,r1,r2,U,v1,v2) %>%
mutate(across(r1:v2,~round(., digits=8)))
hist = get_T_rne_history(g) %>% filter(t<=100)
find.eq.action.cycles(g)
res = study_convergence(g, eq_li = hist)
# Solve the truncated games to check whether we also have an RNE
# with C|C in x0 and D|D in x1.
g = rel_spe(g, delta=0.5, rho=1, r1 = c(2,1,2), r2=c(2,1,2))
spe = get_eq(g)
spe %>%
select(x,ae.lab,r1,r2,U,v1,v2)
# Cycles can also be longer
# CCD CCD
g = g %>% rel_T_rne(T=20, delta=0.9, rho=0.7, save.history = TRUE, tie.breaking = "slack")
get_T_rne_history(g) %>% filter(x=="x0", t <= 10)
# Here CDDD CDDD
g = g %>% rel_T_rne(T=20, delta=0.35, rho=1, save.history = TRUE, tie.breaking = "slack")
get_T_rne_history(g) %>% filter(x=="x0", t <= 10)
# For T = 50 we are already pretty close to this result
get_eq(g)
# We also see how with increasing T we converge towards
# this payoff.
animate_capped_rne_history(g,x=NULL)
# However, there will be a catch because we only
# solve the game numerical with imprecise floating
# point arithmethics...
# Solve for T=500
g = g %>% rel_T_rne(T=500, save.history = TRUE, tie.breaking = "slack")
# hist will contain for all values from T from 1 to 500
# the corresponding RNE payoffs
hist = get_T_rne_history(g) %>%
filter(x=="x0") %>%
mutate(T = 500-t+1)
# Plot r1 as function of T
library(ggplot2)
ggplot(hist, aes(x=t, y=r1)) + geom_point()
# We see that for T slightly above 200
# again high levels of r1 can be implemented
# That is because incentives constraints
# are only checked approximately.
#
# This means at some point it will become
# numerically again incentive compatible to
# implement the punishment action.
#
# If we could exactly compute equilibria
# payoffs would converge to (0,1) without those
# jumps, however.
}
```

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