12 Application: Strategy frequency estimation with a mixed strategy

In the previous chapter, we learned about the strategy frequency estimation method (SFEM). One omission in this chapter was the possibility for including mixed or behavior strategies in the estimation. That is, the previous chapter only coverered strategies that were pure strategies. While the extension to mixing is not difficult, it is worth some attention. In the example below, we are interested in the importance of an equilibrium mixed strategy, compared to some other plausible candidate strategies that players could be using.

12.1 Example dataset and strategies

Anwar and Georgalos (2024) study a sequential 4-player public goods game with poisition uncertainty. That is, participants in this game played sequentially, but did not know exactly the position out of four that they would take their turn. Each player could invest or not invest a fixed amount in a public good (i.e. investment was a binary decision, not continuous). Players receive partial information about the investments of players before them. Specifically,

1. In Treatment 1 players received information about the investment choices of the two players who moved before them. Players did not know their position in the game in this treatment, but could infer whether they were in Position 1 or 2 if they receivec feedback about fewer than 2 opponents.
2. In Treatment 2 players received information about the investment choice of the one player before them. Players did not know their position in the game, but could infer that they were in Position 1 because they would not have received any feedback about opponents’ cholices, and
3. Treatment 3 was the same as Treatment 1, except that players knew their position in the game.

In their SFEM, Anwar and Georgalos (2024) consider four strategies shown in Table 12.1. The “Free rider” and “Altruist” strategeis do not depend on the treatment, and just involve either always not cooperating or cooperating, respectively. The two other strategies, “G&M” and “Conditional cooperator” depend on the treatment. Here the “G&M” strategy is the equilibrium strategy characterized by Gallice and Monzón (2019) for this game. Importantly for this application, in Treatment 2 the strategy invests with probability \(\gamma=0.528\) after some information sets. It is this strategy that will add the novel part to this SFEM compared to the previous chapter.

TAB<-rbind(
  c("G&M, T1", 1,0,1,0,0,1),
  c("G&M, T2", 1,"$\\gamma$",1 ,"$\\gamma$",1," "),
  c("G&M, T3", 0, 0, 0, 0, 0, 0),
  c("Free rider",0,0,0,0,0,0),
  c("Altruist",1,1,1,1,1,1),
  c("Conditional co-operator, T1 & T3",1,0,1,0,1,1),
  c("Conditional co-operator, T2",1, 0, 1, 0, 1, " ")
)

TAB |>
  kbl(caption = "The four strategies considered in @Anwar2024. $\\gamma=0.528$. $c_0$ ($c_1$, $c_2$) indicates the information set where 0 (1, 2) previous players have chosen to invest.") |>
  kable_classic(full_width=FALSE) |>
  add_header_above(c("Strategy","$c_0$","$c_0$","$c_1$","$c_0$","$c_1$","$c_2$")) |>
  add_header_above(c(" ","Position 1","Position 2"=2,"Position 3 & 4"=3)) |>
  add_header_above(c(" ","Information set" = 6))

Table 12.1: The four strategies considered in Anwar and Georgalos (2024). \(\gamma=0.528\). \(c_0\) (\(c_1\), \(c_2\)) indicates the information set where 0 (1, 2) previous players have chosen to invest.

	Information set
	Position 1	Position 2		Position 3 & 4
Strategy	\(c_0\)	\(c_0\)	\(c_1\)	\(c_0\)	\(c_1\)	\(c_2\)
G&M, T1	1	0	1	0	0	1
G&M, T2	1	\(\gamma\)	1	\(\gamma\)	1
G&M, T3	0	0	0	0	0	0
Free rider	0	0	0	0	0	0
Altruist	1	1	1	1	1	1
Conditional co-operator, T1 & T3	1	0	1	0	1	1
Conditional co-operator, T2	1	0	1	0	1

12.2 The likelihood function

Note that we can write each strategy as a 6-vector, where each element represents the probability of playing “invest” in each information set. That is, the G&M strategy for Treatment 1 can be represented by:

\[ \sigma_s = \begin{pmatrix} 1&0&1&0&0&1 \end{pmatrix}^\top \]

and the G&M strategy for Treatment 2 can be represented by:

\[ \sigma_s=\begin{pmatrix} 1&0.528 & 1 & 0.528 & 1& 0.5 \end{pmatrix} \] Here since the “Position 3 & 4, \(c_2\)” information set is never reached, we can in principle put anything into this element of the vector. However since we will be taking the log of these vectors to calculate the log-likelihood, 0.5 will mean we don’t end up taking the log of a negative number.

Since there are still some (nearly all) pure strategies on the list, we still need the tremble probability \(\epsilon\in(0,0.5)\) for these pure strategies to make sure that the likelihood function is not zero everywhere. Therefore we can define the strategy inclusive of trembles as the probability a player chooses “invest” after each of the six information sets:

\[ \tilde\sigma_s=(1-\epsilon)\sigma_s+\epsilon(1-\sigma_s) \]

The only bits we need to worry about is for the second treatment, where the G&M strategy is mixed. Here we just replace the relevant elements of \(\tilde\sigma_s\) with \(\gamma\).⁵⁵ That is, the G&M strategy in Treatment 2 becomes:

\[ \tilde\sigma_{\text{G&M, T2}}=\begin{pmatrix} 1-\epsilon & 0.528 & 1-\epsilon & 0.528 & 1-\epsilon & 0.5 \end{pmatrix}^\top \]

Now we can construct the log-likelihood of following a particular strategy \(\sigma_s\) for a participant \(i\):

\[ \log p(y_i,n_i\mid \sigma_s,\epsilon)=y_i^\top \log(\tilde\sigma_s)+(n_i-y_i)^\top \log(1-\tilde\sigma_s) \]

Where \(y_i\) is a count of the number of times participant \(i\) chose “invest” after each information set, and \(n_i\) is a count of the number of times participant \(i\) reached each information set.

The rest proceeds exactly as the SFEM would in the previous chapter. We next take the mixing probabilities \(\psi\in\Delta^4\) and integrate out the types:

\[ \log p(y_i,n_i\mid \psi,\epsilon)=\log\left(\sum_{s=1}^4\exp\left(\log\psi_s+\log p(y_i,n_i\mid \sigma_s,\epsilon)\right)\right) \]

Note, as we did in the previous chapter, that we are writing this in a form that means we can use Stan’s log_sum_exp() function.

12.3 Implementation in Stan

The parameters in the model are the strategy frequencies \(\psi\in \Delta^4\), and the tremble prbability \(\epsilon\in(0,0.5)\). I assign the following priors to these parameters:

\[ \begin{aligned} \psi&\sim\mathrm{Dirichlet}(1,1,1,1)\\ \epsilon&\sim\mathrm{TruncatedBeta}(1,1,(0,0.5)) \end{aligned} \]

I estimate five models that make different assumptions about how the strategies and trembles relate between treatments. Because all of these programs can use the same data and transformed data blocks, I wrote a single script called SFEM_INCLUDE_data.stan to include at the top of alll the Stan programs. Here is SFEM_INCLUDE_data.stan:

data {
  
  //
  real<lower=0,upper=1> gamma;
  
  int n_t1;
  matrix[n_t1,6] coop_t1;
  matrix[n_t1,6] count_t1;
  int n_t2;
  matrix[n_t2,6] coop_t2;
  matrix[n_t2,6] count_t2;
  int n_t3;
  matrix[n_t3,6] coop_t3;
  matrix[n_t3,6] count_t3;
  
  
  vector[4] prior_mix;
  vector[2] prior_eps;

}

transformed data {
  
  // description of strategies. See Table 3 in the paper
  
  row_vector[6] strg_t1[4];
  row_vector[6] strg_t2[4];
  row_vector[6] strg_t3[4];
  
  // G&M
  strg_t1[1] = [1,0,1,0,0,1];
  strg_t2[1] = [1,gamma,1,gamma,1,0.5];
  strg_t3[1] = rep_row_vector(0,6);
  
  // free rider
  strg_t1[2] = rep_row_vector(0,6);
  strg_t2[2] = rep_row_vector(0,6);
  strg_t3[2] = rep_row_vector(0,6);
  
  // altruist
  strg_t1[3] = rep_row_vector(1,6);
  strg_t2[3] = rep_row_vector(1,6);
  strg_t3[3] = rep_row_vector(1,6);
  
  // conditional cooperator
  strg_t1[4] = [1,0,1,0,1,1];
  strg_t2[4] = [1,0,1,0,1,0.5];
  strg_t3[4] = [1,0,1,0,1,1];
  
}

First, I estimate a model that allows the strategy frequencies \(\psi\) and the tremble probability \(\epsilon\) to vary by treatment. This is probably the best comparison to the SFEM estimated in Anwar and Georgalos (2024) (see their Table 4):

#include SFEM_INCLUDE_data.stan

parameters {
  
  simplex[4] mix[3];
  
  vector<lower=0,upper=0.5>[3] eps;

}

model {
  
  // get a matrix of cooperation probabilities for each strategy in each treatment
  
  vector[6] pr_coop_t1[4];
  vector[6] pr_coop_t2[4];
  vector[6] pr_coop_t3[4];
  for (ss in 1:4) {
    

    
    pr_coop_t1[ss] = strg_t1[ss]'*(1-eps[1])+(1-strg_t1[ss])'*eps[1];
    pr_coop_t2[ss] = strg_t2[ss]'*(1-eps[2])+(1-strg_t2[ss])'*eps[2];
    pr_coop_t3[ss] = strg_t3[ss]'*(1-eps[3])+(1-strg_t3[ss])'*eps[3];
    
    // fix the strategies that are actually mixed.
    for (tt in 1:6) {
      
      pr_coop_t1[ss][tt] = abs(strg_t1[ss][tt]-0.5)<0.4 ? gamma : pr_coop_t1[ss][tt];
      pr_coop_t2[ss][tt] = abs(strg_t2[ss][tt]-0.5)<0.4 ? gamma : pr_coop_t2[ss][tt];
      pr_coop_t3[ss][tt] = abs(strg_t3[ss][tt]-0.5)<0.4 ? gamma : pr_coop_t3[ss][tt];
    }
    
  }
  
  matrix[n_t1,4] like_t1 = rep_matrix(log(mix[1]'),n_t1);
  matrix[n_t2,4] like_t2 = rep_matrix(log(mix[2]'),n_t2);
  matrix[n_t3,4] like_t3 = rep_matrix(log(mix[3]'),n_t3);
  
  for (ss in 1:4) {
    
    like_t1[,ss]+=coop_t1*log(pr_coop_t1[ss])+(count_t1-coop_t1)*log(1-pr_coop_t1[ss]);
    like_t2[,ss]+=coop_t2*log(pr_coop_t2[ss])+(count_t2-coop_t2)*log(1-pr_coop_t2[ss]);
    like_t3[,ss]+=coop_t3*log(pr_coop_t3[ss])+(count_t3-coop_t3)*log(1-pr_coop_t3[ss]);

  }
  
  for (ii in 1:n_t1) {
    target += log_sum_exp(like_t1[ii,]);
  }
  for (ii in 1:n_t2) {
    target += log_sum_exp(like_t2[ii,]);
  }
  for (ii in 1:n_t3) {
    target += log_sum_exp(like_t3[ii,]);
  }
  
  for (tt in 1:3) {
    
    target += dirichlet_lpdf(mix[tt]|prior_mix);
    target += beta_lpdf(eps[tt]| prior_eps[1],prior_eps[2]);
    
  }

}

However what I was interested in (and this is not necessarily what Anwar and Georgalos (2024) were interested in) was how stable the strategy frequencies were across treatments, so I estimated a model that assumed that the strategy frequencies were the same across the three treatments, but allowed the tremble probability to vary across treatments:

#include SFEM_INCLUDE_data.stan

parameters {
  
  simplex[4] mix;
  
  vector<lower=0,upper=0.5>[3] eps;

}

model {
  
  // get a matrix of cooperation probabilities for each strategy in each treatment
  
  vector[6] pr_coop_t1[4];
  vector[6] pr_coop_t2[4];
  vector[6] pr_coop_t3[4];
  for (ss in 1:4) {
    
    pr_coop_t1[ss] = strg_t1[ss]'*(1-eps[1])+(1-strg_t1[ss])'*eps[1];
    pr_coop_t2[ss] = strg_t2[ss]'*(1-eps[2])+(1-strg_t2[ss])'*eps[2];
    pr_coop_t3[ss] = strg_t3[ss]'*(1-eps[3])+(1-strg_t3[ss])'*eps[3];
    
    // fix the strategies that are actually mixed.
    for (tt in 1:6) {
      
      pr_coop_t1[ss][tt] = abs(strg_t1[ss][tt]-0.5)<0.4 ? gamma : pr_coop_t1[ss][tt];
      pr_coop_t2[ss][tt] = abs(strg_t2[ss][tt]-0.5)<0.4 ? gamma : pr_coop_t2[ss][tt];
      pr_coop_t3[ss][tt] = abs(strg_t3[ss][tt]-0.5)<0.4 ? gamma : pr_coop_t3[ss][tt];
    }
    
  }
  
  matrix[n_t1,4] like_t1 = rep_matrix(log(mix'),n_t1);
  matrix[n_t2,4] like_t2 = rep_matrix(log(mix'),n_t2);
  matrix[n_t3,4] like_t3 = rep_matrix(log(mix'),n_t3);
  
  for (ss in 1:4) {
    
    like_t1[,ss]+=coop_t1*log(pr_coop_t1[ss])+(count_t1-coop_t1)*log(1-pr_coop_t1[ss]);
    like_t2[,ss]+=coop_t2*log(pr_coop_t2[ss])+(count_t2-coop_t2)*log(1-pr_coop_t2[ss]);
    like_t3[,ss]+=coop_t3*log(pr_coop_t3[ss])+(count_t3-coop_t3)*log(1-pr_coop_t3[ss]);

  }
  
  for (ii in 1:n_t1) {
    target += log_sum_exp(like_t1[ii,]);
  }
  for (ii in 1:n_t2) {
    target += log_sum_exp(like_t2[ii,]);
  }
  for (ii in 1:n_t3) {
    target += log_sum_exp(like_t3[ii,]);
  }
  
  for (tt in 1:3) {
    
    
    target += beta_lpdf(eps[tt]| prior_eps[1],prior_eps[2]);
    
  }
  
  target += dirichlet_lpdf(mix|prior_mix);

}

For good measure, I also estimated a model that imposed that the tremble probabilities were constant across the treatments:

#include SFEM_INCLUDE_data.stan

parameters {
  
  simplex[4] mix;
  
  vector<lower=0,upper=0.5>[3] eps;

}

model {
  
  // get a matrix of cooperation probabilities for each strategy in each treatment
  
  vector[6] pr_coop_t1[4];
  vector[6] pr_coop_t2[4];
  vector[6] pr_coop_t3[4];
  for (ss in 1:4) {
    
    pr_coop_t1[ss] = strg_t1[ss]'*(1-eps[1])+(1-strg_t1[ss])'*eps[1];
    pr_coop_t2[ss] = strg_t2[ss]'*(1-eps[2])+(1-strg_t2[ss])'*eps[2];
    pr_coop_t3[ss] = strg_t3[ss]'*(1-eps[3])+(1-strg_t3[ss])'*eps[3];
    
    // fix the strategies that are actually mixed.
    for (tt in 1:6) {
      
      pr_coop_t1[ss][tt] = abs(strg_t1[ss][tt]-0.5)<0.4 ? gamma : pr_coop_t1[ss][tt];
      pr_coop_t2[ss][tt] = abs(strg_t2[ss][tt]-0.5)<0.4 ? gamma : pr_coop_t2[ss][tt];
      pr_coop_t3[ss][tt] = abs(strg_t3[ss][tt]-0.5)<0.4 ? gamma : pr_coop_t3[ss][tt];
    }
    
  }
  
  matrix[n_t1,4] like_t1 = rep_matrix(log(mix'),n_t1);
  matrix[n_t2,4] like_t2 = rep_matrix(log(mix'),n_t2);
  matrix[n_t3,4] like_t3 = rep_matrix(log(mix'),n_t3);
  
  for (ss in 1:4) {
    
    like_t1[,ss]+=coop_t1*log(pr_coop_t1[ss])+(count_t1-coop_t1)*log(1-pr_coop_t1[ss]);
    like_t2[,ss]+=coop_t2*log(pr_coop_t2[ss])+(count_t2-coop_t2)*log(1-pr_coop_t2[ss]);
    like_t3[,ss]+=coop_t3*log(pr_coop_t3[ss])+(count_t3-coop_t3)*log(1-pr_coop_t3[ss]);

  }
  
  for (ii in 1:n_t1) {
    target += log_sum_exp(like_t1[ii,]);
  }
  for (ii in 1:n_t2) {
    target += log_sum_exp(like_t2[ii,]);
  }
  for (ii in 1:n_t3) {
    target += log_sum_exp(like_t3[ii,]);
  }
  
  for (tt in 1:3) {
    
    
    target += beta_lpdf(eps[tt]| prior_eps[1],prior_eps[2]);
    
  }
  
  target += dirichlet_lpdf(mix|prior_mix);

}

To test whether all players were using the G&M strategy, I estimated a SFEM with just the G&M strategy. Comparing this to the others would give us an idea of how important the free rider, altruist, and conditional co-operator stratgies were:

#include SFEM_INCLUDE_data.stan

parameters {
  
  vector<lower=0,upper=0.5>[3] eps;

}

model {
  
  // get a matrix of cooperation probabilities for each strategy in each treatment
  
  vector[6] pr_coop_t1[4];
  vector[6] pr_coop_t2[4];
  vector[6] pr_coop_t3[4];
  for (ss in 1:4) {
    
    pr_coop_t1[ss] = strg_t1[ss]'*(1-eps[1])+(1-strg_t1[ss])'*eps[1];
    pr_coop_t2[ss] = strg_t2[ss]'*(1-eps[2])+(1-strg_t2[ss])'*eps[2];
    pr_coop_t3[ss] = strg_t3[ss]'*(1-eps[3])+(1-strg_t3[ss])'*eps[3];
    
    // fix the strategies that are actually mixed.
    for (tt in 1:6) {
      
      pr_coop_t1[ss][tt] = abs(strg_t1[ss][tt]-0.5)<0.4 ? gamma : pr_coop_t1[ss][tt];
      pr_coop_t2[ss][tt] = abs(strg_t2[ss][tt]-0.5)<0.4 ? gamma : pr_coop_t2[ss][tt];
      pr_coop_t3[ss][tt] = abs(strg_t3[ss][tt]-0.5)<0.4 ? gamma : pr_coop_t3[ss][tt];
    }
    
  }
  
  matrix[n_t1,1] like_t1 = rep_matrix(0,n_t1,1);
  matrix[n_t2,1] like_t2 = rep_matrix(0,n_t2,1);
  matrix[n_t3,1] like_t3 = rep_matrix(0,n_t3,1);
  
  for (ss in 1:1) {
    
    like_t1[,ss]+=coop_t1*log(pr_coop_t1[ss])+(count_t1-coop_t1)*log(1-pr_coop_t1[ss]);
    like_t2[,ss]+=coop_t2*log(pr_coop_t2[ss])+(count_t2-coop_t2)*log(1-pr_coop_t2[ss]);
    like_t3[,ss]+=coop_t3*log(pr_coop_t3[ss])+(count_t3-coop_t3)*log(1-pr_coop_t3[ss]);

  }
  
  for (ii in 1:n_t1) {
    target += like_t1[ii,1];
  }
  for (ii in 1:n_t2) {
    target += like_t2[ii,1];
  }
  for (ii in 1:n_t3) {
    target += like_t3[ii,1];
  }
  
  for (tt in 1:3) {
    
    
    target += beta_lpdf(eps[tt]| prior_eps[1],prior_eps[2]);
    
  }
  

}

Finally, to test the importance of the G&M strategy, I estimated a model without this strategy. That is, if this strategy is not important, then eliminating it from the list should not matter too much:

#include SFEM_INCLUDE_data.stan

parameters {
  
  simplex[3] mix[3];
  
  vector<lower=0,upper=0.5>[3] eps;

}

model {
  
  // get a matrix of cooperation probabilities for each strategy in each treatment
  
  vector[6] pr_coop_t1[3];
  vector[6] pr_coop_t2[3];
  vector[6] pr_coop_t3[3];
  for (ss in 1:3) {
    
    pr_coop_t1[ss] = strg_t1[ss+1]'*(1-eps[1])+(1-strg_t1[ss+1])'*eps[1];
    pr_coop_t2[ss] = strg_t2[ss+1]'*(1-eps[2])+(1-strg_t2[ss+1])'*eps[2];
    pr_coop_t3[ss] = strg_t3[ss+1]'*(1-eps[3])+(1-strg_t3[ss+1])'*eps[3];
    
    // fix the strategies that are actually mixed.
    for (tt in 1:6) {
      
      pr_coop_t1[ss][tt] = abs(strg_t1[ss][tt]-0.5)<0.4 ? gamma : pr_coop_t1[ss][tt];
      pr_coop_t2[ss][tt] = abs(strg_t2[ss][tt]-0.5)<0.4 ? gamma : pr_coop_t2[ss][tt];
      pr_coop_t3[ss][tt] = abs(strg_t3[ss][tt]-0.5)<0.4 ? gamma : pr_coop_t3[ss][tt];
    }
    
  }
  
  matrix[n_t1,3] like_t1 = rep_matrix(log(mix[1]'),n_t1);
  matrix[n_t2,3] like_t2 = rep_matrix(log(mix[2]'),n_t2);
  matrix[n_t3,3] like_t3 = rep_matrix(log(mix[3]'),n_t3);
  
  for (ss in 1:3) {
    
    like_t1[,ss]+=coop_t1*log(pr_coop_t1[ss])+(count_t1-coop_t1)*log(1-pr_coop_t1[ss]);
    like_t2[,ss]+=coop_t2*log(pr_coop_t2[ss])+(count_t2-coop_t2)*log(1-pr_coop_t2[ss]);
    like_t3[,ss]+=coop_t3*log(pr_coop_t3[ss])+(count_t3-coop_t3)*log(1-pr_coop_t3[ss]);
    
    

  }
  
  for (ii in 1:n_t1) {
    target += log_sum_exp(like_t1[ii,]);
  }
  for (ii in 1:n_t2) {
    target += log_sum_exp(like_t2[ii,]);
  }
  for (ii in 1:n_t3) {
    target += log_sum_exp(like_t3[ii,]);
  }
  
  for (tt in 1:3) {
    
    target += dirichlet_lpdf(mix[tt]|prior_mix[1:3]);
    target += beta_lpdf(eps[tt]| prior_eps[1],prior_eps[2]);
    
  }

}

12.4 Results

Table 12.2 shows the posterior estimates from the most general model, where strategy frequencies and tremble probabilities can vary by treatment. Here we can see that the G&M strategy accounts for a substantial fraction of decisions in all three treatments. For perspective with the Dirichlet prior, the prior stanard deviation for the mixing probabilities is about 0.19, which is quite a bit larget than the posterior standard deviations: we have learned substantially from the data.

fmt<-"%.3f"

strgList<-c("G&M","Free rider","Altruist","Conditional cooperator")

Fit<-summary("Code/Anwar2024/Fit_SFEM.rds" |> readRDS())$summary |>
  data.frame() |>
  rownames_to_column(var = "par") |>
  filter(par!="lp__") |>
  mutate(
    parameter = ifelse(grepl("mix",par),"mix","epsilon")
  ) |>
  mutate(
    treatment = ifelse(parameter=="mix",
                       str_split_i(par,",",1) |> parse_number(),
                       par |> parse_number()
                       ),
    strg = str_split_i(par,",",2) |> parse_number() ,
    strategy = ifelse(parameter=="mix",strgList[strg],"$\\epsilon$")
  ) 

means<-Fit |>
  mutate(
    mean = sprintf(fmt,mean)
  ) |>
  pivot_wider(
    id_cols = "strategy",
    names_from = "treatment",
    values_from = "mean"
  )

sds<-Fit |>
  mutate(
    sd = paste0("(",sprintf(fmt,sd),")")
  ) |>
  pivot_wider(
    id_cols = "strategy",
    names_from = "treatment",
    values_from = "sd"
  ) |>
  mutate(
    strategy = ""
  )

TAB<-tibble()

for (rr in 1:dim(means)[1]) {
  
  TAB<-rbind(TAB,
             means[rr,],
             sds[rr,],
             rep("  ",4)
             )
}

TAB |>
  kbl(caption = "Strategy frequencies and tremble probabilities from the most general model allowing for strategy frequencies and tremble probabilities to vary by treatment. ") |>
  kable_classic(full_width=FALSE) |>
  add_header_above(c("","Treatment"=3))

Table 12.2: Strategy frequencies and tremble probabilities from the most general model allowing for strategy frequencies and tremble probabilities to vary by treatment.

	Treatment
strategy	1	2	3
G&M	0.186	0.252	0.142
	(0.069)	(0.091)	(0.093)
Free rider	0.083	0.112	0.144
	(0.045)	(0.051)	(0.092)
Altruist	0.312	0.392	0.097
	(0.079)	(0.091)	(0.056)
Conditional cooperator	0.419	0.244	0.616
	(0.088)	(0.078)	(0.087)
\(\epsilon\)	0.136	0.109	0.182
	(0.013)	(0.015)	(0.014)

In Table 12.3 I show the five models’ posterior probabilities assuming equal prior probabilities. Here we can see most starkly that the final two models do not organize the data well relative to the others. Specifically (i) the G&M strategy on its own does not do well, and (ii) eliminating the G&M strategy only does not do well. Of the remaining models, the most general does the best, but the support for this unrestricted model is not overwhelming. It seems that assuming that strategy frequencies are common across all thre treatments is not too terrible of a restriction.

postprobs<-"Code/Anwar2024/postprobs.rds"|> readRDS()

TAB<-tibble(
  model = c("Unrestricted","Pooled frequencies","Pooled frequencies and trembles","G&M strategy only","Excluding G&M strategy"),
  `posterior probability` = postprobs
)

TAB |>
  kbl(digits=3,caption = "Model posterior probabilities assuming equal prior probabilities. ") |>
  kable_classic(full_width=FALSE)

Table 12.3: Model posterior probabilities assuming equal prior probabilities.

model	posterior probability
Unrestricted	0.548
Pooled frequencies	0.204
Pooled frequencies and trembles	0.247
G&M strategy only	0.000
Excluding G&M strategy	0.000

12.5 R code used to estimate the models

library(tidyverse)
library(rstan)
options(mc.cores = parallel::detectCores())
rstan_options(auto_write = TRUE)
library(bridgesampling)

#------------------------------------------------------------------------------
# Here I am following "SFEM.R" in the replication files to wrangle the data

data<-"Data/AG2024PositionUncertainty.csv" |>
  read.csv()
dt2=data[data$treatment==2,]
dt3=data[data$treatment==3,]
dt4=data[data$treatment==4,]

t2=c()
#Order the data per treatment, player, round and position.
for (round in 1:10){
  position=dt2[,which(colnames(dt2)==paste("pg_T2.",round,".player.id_in_group",sep=""))]
  c0p1=dt2[,which(colnames(dt2)==paste("pg_T2.",round,".player.response_0p1",sep=""))]
  c0p2=dt2[,which(colnames(dt2)==paste("pg_T2.",round,".player.response_0p2",sep=""))]
  c0p34=dt2[,which(colnames(dt2)==paste("pg_T2.",round,".player.response_0p34",sep=""))]
  c1p2=dt2[,which(colnames(dt2)==paste("pg_T2.",round,".player.response_1p2",sep=""))]
  c1p34=dt2[,which(colnames(dt2)==paste("pg_T2.",round,".player.response_1p34",sep=""))]  
  c2p34=dt2[,which(colnames(dt2)==paste("pg_T2.",round,".player.response_2p34",sep=""))]
  
  
  t2=rbind(t2,cbind(sbj=dt2$sbj,treatment=rep(2,32),position,rnd=rep(round,32),c0p1,c0p2,c0p34,c1p2,c1p34,c2p34))
}



t3=c()
#Order the data per treatment, player, round and position.
for (round in 1:10){
  position=dt3[,which(colnames(dt3)==paste("pg_T3.",round,".player.id_in_group",sep=""))]
  c0p1=dt3[,which(colnames(dt3)==paste("pg_T3.",round,".player.response_0p1",sep=""))]
  c0p2=dt3[,which(colnames(dt3)==paste("pg_T3.",round,".player.response_0p2",sep=""))]
  c0p34=dt3[,which(colnames(dt3)==paste("pg_T3.",round,".player.response_0p34",sep=""))]
  c1p2=dt3[,which(colnames(dt3)==paste("pg_T3.",round,".player.response_1p2",sep=""))]
  c1p34=dt3[,which(colnames(dt3)==paste("pg_T3.",round,".player.response_1p34",sep=""))] 
  
  t3=rbind(t3,cbind(sbj=dt3$sbj,treatment=rep(3,32),position,rnd=rep(round,32),c0p1,c0p2,c0p34,c1p2,c1p34))
}



t4=c()
#Order the data per treatment, player, round and position.
for (round in 1:10){
  position=dt4[,which(colnames(dt4)==paste("pg_T4.",round,".player.id_in_group",sep=""))]
  c0p1=dt4[,which(colnames(dt4)==paste("pg_T4.",round,".player.response_0p1",sep=""))]
  c0p2=dt4[,which(colnames(dt4)==paste("pg_T4.",round,".player.response_0p2",sep=""))]
  c0p34=dt4[,which(colnames(dt4)==paste("pg_T4.",round,".player.response_0p34",sep=""))]
  c1p2=dt4[,which(colnames(dt4)==paste("pg_T4.",round,".player.response_1p2",sep=""))]
  c1p34=dt4[,which(colnames(dt4)==paste("pg_T4.",round,".player.response_1p34",sep=""))]  
  c2p34=dt4[,which(colnames(dt4)==paste("pg_T4.",round,".player.response_2p34",sep=""))]
  
  t4=rbind(t4,cbind(sbj=dt4$sbj,treatment=rep(4,32),position,rnd=rep(round,32),c0p1,c0p2,c0p34,c1p2,c1p34,c2p34))
}


t2=data.frame(t2)
t3=data.frame(t3)
t4=data.frame(t4)
sbj2=unique(t2$sbj)
sbj3=unique(t3$sbj)
sbj4=unique(t4$sbj)

#-------------------------------------------------------------------------------
# JB's code starts here
#-------------------------------------------------------------------------------

# Get data into long summarized form

D<-t2 |>
  pivot_longer(cols = c0p1:c2p34) |>
  rbind(
    t3 |>
      pivot_longer(cols = c0p1:c1p34)
  ) |>
  rbind(
    t4 |>
      pivot_longer(cols = c0p1:c2p34)
  ) |>
  mutate(
    coop = value==10
  ) |>
  filter(!is.na(coop)) |>
  select(-value) |>
  mutate(
    pos = str_split_i(name,"p",2),
    history_length = ifelse(pos=="1",0,nchar(pos)),
    history_coop = str_split_i(name,"p",1) |> parse_number()
  ) |>
  group_by(sbj,treatment,history_length,history_coop) |>
  summarize(
    coop_count = sum(coop),
    action_count = n()
  ) |>
  mutate(
    treatment = treatment-1,
    ps = paste0("p",history_length+1,"c",history_coop)
  )

dwide<- D |>
  pivot_wider(
    id_cols = c(sbj,treatment),
    names_from = ps,
    values_from = c(coop_count,action_count),
    values_fill = 0
  ) |> 
  ungroup()

t1<-dwide |> filter(treatment==1)
coop_t1<-t1 |> select(contains("coop_count"))
count_t1<-t1 |> select(contains("action_count"))

t2<-dwide |> filter(treatment==2)
coop_t2<-t2 |> select(contains("coop_count"))
count_t2<-t2 |> select(contains("action_count"))

t3<-dwide |> filter(treatment==3)
coop_t3<-t3 |> select(contains("coop_count"))
count_t3<-t3 |> select(contains("action_count"))




#-------------------------------------------------------------------------------

model<-"Code/Anwar2024/SFEM.stan" |>
  stan_model()

dStan<-list(
  
  gamma = 0.528,
  
  n_t1 = dim(coop_t1)[1],
  coop_t1 = coop_t1,
  count_t1 = count_t1,
  
  n_t2 = dim(coop_t2)[1],
  coop_t2 = coop_t2,
  count_t2 = count_t2,
  
  n_t3 = dim(coop_t3)[1],
  coop_t3 = coop_t3,
  count_t3 = count_t3,
  
  prior_mix = c(1,1,1,1),
  prior_eps = c(1,1)
)

Fit<-model |>
  sampling(data=dStan,seed=42)



model_pooledfreq<-"Code/Anwar2024/SFEM_pooledfrequencies.stan" |>
  stan_model()

Fit_pooledfreq<-model_pooledfreq |>
  sampling(data=dStan,seed=42)

model_pooledall<-"Code/Anwar2024/SFEM_pooledall.stan" |>
  stan_model()

Fit_pooledall<-model_pooledall |>
  sampling(data=dStan,seed=42)

model_strg1only<-"Code/Anwar2024/SFEM_strg1only.stan" |>
  stan_model()

Fit_strg1only<-model_strg1only |>
  sampling(data=dStan,seed=42)

model_notGM<-"Code/Anwar2024/SFEM_notGM.stan" |>
  stan_model()

Fit_notGM<-model_notGM |>
  sampling(data=dStan,seed=42)

bs_unrestricted<-Fit |> bridge_sampler()
bs_pooledfreq<-Fit_pooledfreq |> bridge_sampler()
bs_pooledall<-Fit_pooledall |> bridge_sampler()
bs_strg1only<-Fit_strg1only |> bridge_sampler()
bs_notGM<-Fit_notGM |> bridge_sampler()

Fit |>
  saveRDS("Code/Anwar2024/Fit_SFEM.rds")


(postprobs<-post_prob(bs_unrestricted,bs_pooledfreq,bs_pooledall,bs_strg1only,bs_notGM))

postprobs |>
  saveRDS("Code/Anwar2024/postprobs.rds")

References

Anwar, Chowdhury Mohammad Sakib, and Konstantinos Georgalos. 2024. “Position Uncertainty in a Sequential Public Goods Game: An Experiment.” Experimental Economics, 1–34.

Gallice, Andrea, and Ignacio Monzón. 2019. “Co-Operation in Social Dilemmas Through Position Uncertainty.” The Economic Journal 129 (621): 2137–54.

---

55. This is the same as the original implimentation in Anwar and Georgalos (2024). An alternative implementation could be to keep \(\tilde\sigma_s\) as it is described above. In this case, the participant mixes and trembles. ↩︎