Show code
p_eq_mp
MP — Matching Pennies
GT Behaviour · GTEMO Experiment
1 — Outcomes & P1 Win
Objective
Describe the joint outcomes reached by each couple in Part 1 and Part 2, and test whether the observed distribution deviates from the uniform prediction implied by the mixed-strategy Nash equilibrium (\(p = 0.25\) per outcome).
Outcome distributions
Test: are joint outcomes uniformly distributed?
Show code
tab_obs_exp_mp |>
gt::gt(groupname_col = "Part") |>
gt::tab_header(
title = "MP — Observed vs expected outcome counts",
subtitle = "Expected = N × 0.25 under uniform NE prediction"
) |>
gt::cols_label(Outcome = "Outcome", Observed = "Observed",
Expected = "Expected (NE)", N = "N couples") |>
gt::cols_hide(N) |>
gt::tab_style(style = gt::cell_text(weight = "bold"),
locations = gt::cells_column_labels()) |>
gt::tab_style(style = gt::cell_text(weight = "bold"),
locations = gt::cells_row_groups())| MP — Observed vs expected outcome counts | ||
| Expected = N × 0.25 under uniform NE prediction | ||
| Outcome | Observed | Expected (NE) |
|---|---|---|
| Part 1 | ||
| (D,D) | 3 | 3.75 |
| (D,T) | 6 | 3.75 |
| (T,D) | 3 | 3.75 |
| (T,T) | 3 | 3.75 |
| Part 2 | ||
| (D,D) | 0 | 3.75 |
| (D,T) | 6 | 3.75 |
| (T,D) | 3 | 3.75 |
| (T,T) | 6 | 3.75 |
Show code
tab_chi_uniform_mp |>
gt::gt() |>
gt::tab_header(
title = "MP — χ² goodness-of-fit vs uniform distribution",
subtitle = "H₀: each joint outcome has probability 0.25 (mixed-strategy NE). p-value Monte Carlo (B = 9 999)."
) |>
gt::cols_label(Part = "Phase", N = "N", chi2 = "χ²",
df = "df", p_sim = "p (sim.)", note = "Note") |>
gt::fmt_number(columns = chi2, decimals = 3) |>
gt::tab_style(style = gt::cell_text(weight = "bold"),
locations = gt::cells_body(columns = p_sim,
rows = p_sim < 0.05)) |>
gt::tab_style(style = gt::cell_text(weight = "bold"),
locations = gt::cells_column_labels())| MP — χ² goodness-of-fit vs uniform distribution | |||||
| H₀: each joint outcome has probability 0.25 (mixed-strategy NE). p-value Monte Carlo (B = 9 999). | |||||
| Phase | N | χ² | df | p (sim.) | Note |
|---|---|---|---|---|---|
| Part 1 | 15 | 1.800 | NA | 0.6999 | Expected per cell = 3.75 — p-value Monte Carlo (B = 9 999) |
| Part 2 | 15 | 6.600 | NA | 0.0963 | Expected per cell = 3.75 — p-value Monte Carlo (B = 9 999) |
Conditioning on session gender
Show code
tab_cond_coord_mp |>
gt::gt() |>
gt::tab_header(title = "MP — P1 Win and choice T by session gender",
subtitle = "χ² with Monte Carlo simulated p-value (B = 2000), couple level") |>
gt::cols_label(Outcome = "Outcome", Factor = "Factor", Test = "χ²(sim.) test") |>
gt::tab_style(style = gt::cell_text(weight = "bold"),
locations = gt::cells_column_labels()) |>
gt::opt_stylize(style = 1) |>
gt::tab_options(table.font.size = 13)
p_eq_cond_mp| MP — P1 Win and choice T by session gender | ||
| χ² with Monte Carlo simulated p-value (B = 2000), couple level | ||
| Outcome | Factor | χ²(sim.) test |
|---|---|---|
| Coordination Part 2 | Session gender | χ²(sim.): p = 0.317 ns |
| Coordination Part 1 | Session gender | χ²(sim.): p = 1.000 ns |
| Mutual choice T Part 2 | Session gender | χ²(sim.): p = 0.323 ns |
| Mutual choice T Part 1 | Session gender | χ²(sim.): p = 1.000 ns |
2 — Choice & signal distributions
Objective
Describe the marginal distributions of choices and signals (Part 1 choice → signal sent → Part 2 choice), and examine how the opponent’s signal shapes Part 2 behaviour. All proportions use exact 95% Clopper-Pearson CIs.
Distributions table
Show code
tab_mp| MP — Choice and signal distributions | ||||
| 95% Clopper-Pearson CI | ||||
| Choice / Signal | n | N | % | 95% CI |
|---|---|---|---|---|
| Part 1 | ||||
| T | 15 | 30 | 50.0% | [31.3%, 68.7%] |
| D | 15 | 30 | 50.0% | [31.3%, 68.7%] |
| Signal | ||||
| T | 19 | 30 | 63.3% | [43.9%, 80.1%] |
| D | 11 | 30 | 36.7% | [19.9%, 56.1%] |
| Part 2 | ||||
| T | 21 | 30 | 70.0% | [50.6%, 85.3%] |
| D | 9 | 30 | 30.0% | [14.7%, 49.4%] |
Proportions by phase
Show code
p_mp_dist
Within-subject choice shift: McNemar test (Part 1 vs Part 2)
Show code
tab_mcnemar_mp |>
gt::gt() |>
gt::tab_header(
title = "McNemar test — MP: choice_1_T vs choice_2_T",
subtitle = "Paired within-individual"
) |>
gt::cols_label(statistic = "χ²", p_value = "p-value", n = "n", note = "Note") |>
gt::fmt_number(columns = c(statistic, p_value), decimals = 4) |>
gt::tab_style(style = gt::cell_text(weight = "bold"),
locations = gt::cells_body(columns = p_value,
rows = !is.na(p_value) & p_value < 0.05)) |>
gt::tab_style(style = gt::cell_text(weight = "bold"),
locations = gt::cells_column_labels())| McNemar test — MP: choice_1_T vs choice_2_T | |||
| Paired within-individual | |||
| χ² | p-value | n | Note |
|---|---|---|---|
| 2.5000 | 0.1138 | 30 | OK |
Note
A significant McNemar result (p < 0.05) indicates that pre-play cheap talk systematically shifted individual choice T rates — consistent with cheap talk influencing behaviour even in MP, where theory predicts signals should have no effect because players have strictly opposing interests and no credible commitment is possible.
Opponent signal and the information set before Part 2
Important
Decision sequence. After sending their own signal and before making the Part 2 choice, each player observes the opponent’s signal. The Part 2 decision is taken with a two-dimensional information set: (own signal sent) × (opponent’s signal received). The four possible information sets are: T/T, T/D, D/T, D/D.
Show code
p_sig_heatmap_mp
Show code
p_choice2_by_oppsig_mp
Show code
p_choice2_infoset_mp
Show code
p_follow_opp_mp
Conditioning on gender and role
Show code
tab_cond_sig_mp |>
gt::gt() |>
gt::tab_header(title = "MP — Choice and signal distributions by gender and role",
subtitle = "χ² with Monte Carlo simulated p-value (B = 2000)") |>
gt::cols_label(Outcome = "Outcome", Factor = "Factor", Test = "χ²(sim.) test") |>
gt::tab_style(style = gt::cell_text(weight = "bold"),
locations = gt::cells_column_labels()) |>
gt::opt_stylize(style = 1) |>
gt::tab_options(table.font.size = 13)| MP — Choice and signal distributions by gender and role | ||
| χ² with Monte Carlo simulated p-value (B = 2000) | ||
| Outcome | Factor | χ²(sim.) test |
|---|---|---|
| Part 1 = T | Gender | χ²(sim.): p = 0.714 ns |
| Part 1 = T | Role | χ²(sim.): p = 0.481 ns |
| Signal = T | Gender | χ²(sim.): p = 0.059 ns |
| Signal = T | Role | χ²(sim.): p = 1.000 ns |
| Part 2 = T | Gender | χ²(sim.): p = 0.454 ns |
| Part 2 = T | Role | χ²(sim.): p = 0.415 ns |
Show code
p_cond_sig_mp
3 — Signal honesty & consistency
Objective
Examine whether signals are honest (= same as the action eventually taken in Part 2) and consistent with Part 1 choices. Assess the prevalence of strategy switches between Part 1 and Part 2.
Honesty and consistency proportions
Show code
tab_sec2_mp |>
gt::gt() |>
gt::tab_header(title = "MP — Signal honesty and consistency",
subtitle = "95% Clopper-Pearson CI. Each row is a binary indicator (1 = yes, 0 = no).") |>
gt::cols_label(variable = "Measure", n = "n (=1)", N = "N",
pct = "%", ci95 = "95% CI") |>
gt::tab_style(style = gt::cell_text(weight = "bold"),
locations = gt::cells_column_labels()) |>
gt::tab_footnote(
footnote = "1 if the signal sent equals the Part 2 choice (e.g. sent T and chose T in Part 2). Measures whether players followed through on their signal.",
locations = gt::cells_body(columns = variable, rows = 1)
) |>
gt::tab_footnote(
footnote = "1 if the signal sent equals the Part 1 choice (e.g. signalled T and had also chosen T in Part 1). Measures whether the signal reflects past behaviour — independent of Part 2.",
locations = gt::cells_body(columns = variable, rows = 2)
) |>
gt::tab_footnote(
footnote = "1 if the Part 2 choice differs from the Part 1 choice (choice1 \u2260 choice2). Measures switching behaviour across rounds, independent of the signal. N may differ from rows 1\u20132 due to missing values in different variables.",
locations = gt::cells_body(columns = variable, rows = 3)
) |>
gt::tab_options(table.font.size = 13)| MP — Signal honesty and consistency | ||||
| 95% Clopper-Pearson CI. Each row is a binary indicator (1 = yes, 0 = no). | ||||
| Measure | n (=1) | N | % | 95% CI |
|---|---|---|---|---|
| Signal honest (signal = choice2)1 | 20 | 30 | 66.7% | [47.2%, 82.7%] |
| Signal consistent with Part 1 (signal = choice1)2 | 20 | 30 | 66.7% | [47.2%, 82.7%] |
| Strategy changed Part 1 → Part 2 (choice1 ≠ choice2)3 | 10 | 30 | 33.3% | [17.3%, 52.8%] |
| 1 1 if the signal sent equals the Part 2 choice (e.g. sent T and chose T in Part 2). Measures whether players followed through on their signal. | ||||
| 2 1 if the signal sent equals the Part 1 choice (e.g. signalled T and had also chosen T in Part 1). Measures whether the signal reflects past behaviour — independent of Part 2. | ||||
| 3 1 if the Part 2 choice differs from the Part 1 choice (choice1 ≠ choice2). Measures switching behaviour across rounds, independent of the signal. N may differ from rows 1–2 due to missing values in different variables. | ||||
Binomial test: signal honesty vs 50%
Show code
tab_binom_honest_mp |>
gt::gt() |>
gt::tab_header(title = "Binomial test: P(Honest) vs H₀ = 0.50",
subtitle = "Two-sided test; 95% Clopper-Pearson CI") |>
gt::cols_label(x = "n honest", n = "N", pct = "%", ci95 = "95% CI",
p_value = "p-value") |>
gt::tab_style(style = gt::cell_text(weight = "bold"),
locations = gt::cells_body(columns = p_value,
rows = p_value < 0.05)) |>
gt::tab_style(style = gt::cell_text(weight = "bold"),
locations = gt::cells_column_labels())| Binomial test: P(Honest) vs H₀ = 0.50 | ||||
| Two-sided test; 95% Clopper-Pearson CI | ||||
| n honest | N | % | 95% CI | p-value |
|---|---|---|---|---|
| 20 | 30 | 66.7% | [47.2%, 82.7%] | 0.0987 |
Note
In MP, a signal is honest if the player’s Part 2 action matches what they signalled. Unlike PD, where a D→D case is “trivially honest” because D is dominant, in MP no action is dominant. More importantly, honesty in a zero-sum game has an adverse strategic consequence: a player who honestly signals T and then plays T is revealing their intention to an opponent who benefits from mismatching. Rational players should therefore be strategically deceptive — signal one action and play the other. Observing high honesty rates in MP would indicate that players fail to exploit this deception opportunity, perhaps due to social norms, experimenter demand, or misunderstanding the zero-sum structure.
Strategic transition heatmaps
Show code
p_sankey_mp
Conditioning on gender and role
Show code
tab_cond_honest_mp |>
gt::gt() |>
gt::tab_header(title = "MP — Signal honesty and strategy change by gender and role",
subtitle = "χ² with Monte Carlo simulated p-value (B = 2000)") |>
gt::cols_label(Outcome = "Outcome", Factor = "Factor", Test = "χ²(sim.) test") |>
gt::tab_style(style = gt::cell_text(weight = "bold"),
locations = gt::cells_column_labels()) |>
gt::opt_stylize(style = 1) |>
gt::tab_options(table.font.size = 13)| MP — Signal honesty and strategy change by gender and role | ||
| χ² with Monte Carlo simulated p-value (B = 2000) | ||
| Outcome | Factor | χ²(sim.) test |
|---|---|---|
| Signal honest | Gender | χ²(sim.): p = 1.000 ns |
| Signal honest | Role | χ²(sim.): p = 0.697 ns |
| Strategy changed | Gender | χ²(sim.): p = 1.000 ns |
| Strategy changed | Role | χ²(sim.): p = 1.000 ns |
Show code
p_cond_honest_mp
4 — Belief accuracy & bonus
Objective
Describe the distribution of belief accuracy scores (0 = both beliefs wrong; 1 = one correct; 2 = both correct) and the associated belief bonus payoff. Assess whether belief accuracy is correlated with P1 Win outcomes at the couple level.
Belief accuracy distribution
Show code
p_belief_bar_mp
Hypothesis tests: beliefs, cognitive ability & P1 Win
| MP — Belief accuracy: hypothesis comparison | |||||||||
| H1–H2: individual level. H3: couple level. Stat = Spearman ρ or φ (phi). | |||||||||
| Level | Hypothesis | X | Y | Test | Expected | Stat | p | n | |
|---|---|---|---|---|---|---|---|---|---|
| H1 | Individual | Reflective thinkers (high CRT) predict opponent’s choice more accurately | CRT score (0–4) | Belief accuracy (0–2) | Spearman ρ | Positive | -0.275 | 0.141 | 30 |
| H2 | Individual | Players who made more quiz errors have less accurate beliefs | Quiz errors [log(1+x)] | Belief accuracy (0–2) | Spearman ρ | Negative | 0.046 | 0.809 | 30 |
| H3 | Couple | Couples where both players have perfect beliefs more often result in P1 Win in Part 2 | P1 Win Part 2 (0/1) | Both perfect beliefs (0/1) | Fisher exact + φ1 | Ambiguous | 0.327 | 0.400 | 15 |
| 1 H3 stat = phi coefficient (φ); p-value from two-sided Fisher exact test on 2×2 contingency table. H3 cannot be tested in MP: p1_wins_part2 is not available in the dataset. | |||||||||
Note
H1 tests whether more reflective players (higher CRT) are better at predicting their opponent’s Part 2 choice — if strategic reasoning drives belief formation, a positive Spearman ρ is expected. H2 tests whether players who struggled with game comprehension (more quiz errors) hold less accurate beliefs — expected direction is negative. H3 tests whether couples where both players had perfect beliefs were more likely to result in a P1 Win in Part 2 — the only couple-level hypothesis; uses Fisher exact given small N and binary outcomes. The expected direction for H3 is ambiguous in MP: if P1 correctly predicts P2’s action, P1 can match and win — but if P2 also predicts correctly, P2 can mismatch and P1 loses. Mutual perfect beliefs is self-defeating in a zero-sum game (it would imply a pure-strategy NE, which does not exist in MP). Note: H3 cannot be computed for MP as Part 2 outcome data is not available in the dataset. H1 and H2 are at the individual level; H3 is at the couple level.
Conditioning on gender and role
Show code
p_cond_belief_mp
5 — Econometric models
5.1 — Determinants of belief accuracy
Estimate an ordered logit (MASS::polr) for the belief accuracy score (0 = both beliefs wrong; 1 = one correct; 2 = both correct) across three nested specifications adding Theory of Mind (MASC) and empathy/perspective-taking (IRI) measures. The outcome is treated as an ordered categorical variable: higher scores reflect better first- and second-order belief formation.
WarningSmall-sample caveat
With n = 30 observations and only 6 in the smallest boundary category (min of score 0 and score 2), the events-per-variable (EPV) for M2 is 3 and for M3 is 2 — well below the recommended minimum of 10. polr uses standard MLE and may be biased with small boundary cells. All results are exploratory and should not be used for causal inference.
Model specifications
\[ \begin{aligned} \text{M1:} \quad & \text{logit}\,P(Y \leq j) = \alpha_j - \beta_1\,\text{MASC}_z \\ \text{M2:} \quad & \text{logit}\,P(Y \leq j) = \alpha_j - \beta_1\,\text{MASC}_z - \beta_2\,\text{IRI-PT}_z \\ \text{M3:} \quad & \text{logit}\,P(Y \leq j) = \alpha_j - \beta_1\,\text{MASC}_z - \beta_2\,\text{IRI-PT}_z - \beta_3\,\text{IRI-PD}_z \end{aligned} \]
\(Y\) = belief accuracy score ∈ {0, 1, 2}; \(\alpha_j\) = threshold for category \(j\). MASC = total Theory of Mind score; IRI-PT = Perspective Taking subscale; IRI-PD = Personal Distress subscale. All predictors are z-standardised. OR > 1 indicates a higher probability of achieving a higher accuracy score.
Coefficient table
Show code
tab_olog_gt| MP — Ordered logit: determinants of belief accuracy | |||||||
| DV = belief accuracy score (0/1/2, ordered). n = 30 (score 0: n=6; 1: n=17; 2: n=7). MASC/IRI z-scored. OR > 1 increases probability of higher accuracy.1 | |||||||
| Predictor | β | SE | t | p | OR | OR 2.5% | OR 97.5% |
|---|---|---|---|---|---|---|---|
| M1: MASC only | |||||||
| MASC ToM score (z) | -0.459 | 0.386 | -1.190 | 0.2339 | 0.632 | 0.297 | 1.346 |
| M2: MASC + IRI-PT | |||||||
| MASC ToM score (z) | -0.504 | 0.394 | -1.279 | 0.2010 | 0.604 | 0.268 | 1.289 |
| IRI Perspective Taking (z) | -0.203 | 0.377 | -0.539 | 0.5900 | 0.816 | 0.383 | 1.714 |
| M3: MASC + IRI-PT + IRI-PD | |||||||
| MASC ToM score (z) | -0.432 | 0.408 | -1.059 | 0.2895 | 0.649 | 0.282 | 1.426 |
| IRI Perspective Taking (z) | -0.237 | 0.381 | -0.623 | 0.5335 | 0.789 | 0.366 | 1.666 |
| IRI Personal Distress (z) | -0.272 | 0.375 | -0.726 | 0.4679 | 0.762 | 0.355 | 1.572 |
| 1 Ordered logit (proportional-odds, MASS::polr). p-values from two-tailed z-test on t-statistic. CI from profile likelihood where convergent, otherwise Wald. All models MLE; EPV < 10 — interpret cautiously. | |||||||
Goodness of fit
Show code
tab_gof_olog_gt| MP — Ordered logit: goodness of fit | |||||
| DV = belief accuracy score (0/1/2). EPV = min(n₀, n₂) / k. | |||||
| Model | n | Predictors | EPV | AIC | McFadden R² |
|---|---|---|---|---|---|
| M1: MASC only | 30 | 1 | 6 | 63.5 | 0.0248 |
| M2: MASC + IRI-PT | 30 | 2 | 3 | 65.2 | 0.0297 |
| M3: MASC + IRI-PT + IRI-PD | 30 | 3 | 2 | 66.7 | 0.0388 |
Forest plot: odds ratios
Show code
p_forest_olog
5.2 — Choice T under defection signal
Models
\[ \begin{aligned} \text{M1:} \quad & \text{logit}\,P(T) = \beta_0 + \beta_1\,\text{quiz\_err} \\ \text{M2:} \quad & \text{logit}\,P(T) = \beta_0 + \beta_1\,\text{quiz\_err} + \beta_2\,\text{CRT} \\ \text{M3:} \quad & \text{logit}\,P(T) = \beta_0 + \beta_1\,\text{CRT} \end{aligned} \]
Sample: opp_signal_received = D only. n = 11, events = 6. EPV: M1 = 6, M2 = 3, M3 = 6. All estimated with Firth penalised logit.
Results
| MP — Choice T when opponent signals D | |||||||
| DV = played BR | opp_signal=D. n=11, events=6. EPV: M1=6, M2=3, M3=6. All Firth penalised logit (brglm2).1 | |||||||
| Predictor | β | SE | z | p | OR | OR 2.5% | OR 97.5% |
|---|---|---|---|---|---|---|---|
| M1: Quiz only | |||||||
| Quiz errors [log(1+x)] | -0.052 | 0.559 | -0.09 | 0.926 | 0.95 | 0.32 | 2.84 |
| M2: Quiz + CRT | |||||||
| Quiz errors [log(1+x)] | -0.245 | 0.632 | -0.39 | 0.699 | 0.78 | 0.23 | 2.70 |
| CRT score (0–4) | -1.684 | 1.397 | -1.21 | 0.228 | 0.19 | 0.01 | 2.87 |
| M3: CRT only | |||||||
| CRT score (0–4) | -1.686 | 1.339 | -1.26 | 0.208 | 0.19 | 0.01 | 2.55 |
| 1 Firth penalised logit used for all models (EPV < 10). OR > 1 → increases P(played BR | opp signals D). 95% Wald CI. | |||||||
