Show code
p_eq_bs
BS — Battle of the Sexes
GT Behaviour · GTEMO Experiment
1 — Equilibria & coordination
Objective
Describe the equilibrium outcomes reached by each couple in Part 1 and Part 2, and test whether cheap talk shifted coordination rates using paired McNemar tests.
Equilibrium distributions
Coordination rates: Part 1 vs Part 2
Show code
coord_bs |>
dplyr::select(part, x, n, pct, ci95) |>
gt::gt() |>
gt::tab_header(title = "BS — Coordination rates: Part 1 vs Part 2",
subtitle = "95% Clopper-Pearson CI") |>
gt::cols_label(part = "Phase", x = "n coordinated", n = "N",
pct = "%", ci95 = "95% CI") |>
gt::tab_style(style = gt::cell_text(weight = "bold"),
locations = gt::cells_column_labels())
p_coord_bs| BS — Coordination rates: Part 1 vs Part 2 | ||||
| 95% Clopper-Pearson CI | ||||
| Phase | n coordinated | N | % | 95% CI |
|---|---|---|---|---|
| Part 1 | 6 | 16 | 37.5% | [15.2%, 64.6%] |
| Part 2 | 9 | 16 | 56.2% | [29.9%, 80.2%] |
McNemar tests
Show code
tab_mc_bs |>
gt::gt() |>
gt::tab_header(title = "McNemar tests — BS (couple level)",
subtitle = "Paired Part 1 vs Part 2") |>
gt::cols_label(label = "Test", statistic = "χ²", p_value = "p-value",
note = "Note") |>
gt::fmt_number(columns = c(statistic, p_value), decimals = 4, rows = !is.na(statistic)) |>
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 tests — BS (couple level) | |||
| Paired Part 1 vs Part 2 | |||
| Test | χ² | p-value | Note |
|---|---|---|---|
| BS — coord Part1 vs Part2 | 0.4440 | 0.5050 | OK |
| BS — mutual Demand (D) Part1 vs Part2 | 0.4440 | 0.5050 | OK |
Note
Coordination rate in Part 1: 37.5% of pairs → Part 2: 56.2%. In BS, coordination means landing on either the P1 Preferred (D,T) or P2 Preferred (T,D) equilibrium — both produce positive payoffs (36+12). A significant McNemar result would indicate that cheap talk systematically shifted couples towards coordinated outcomes and away from miscoordination ((D,D) or (T,T), both yielding 0€ for each player).
Conditioning on session gender
Show code
tab_cond_coord_bs |>
gt::gt() |>
gt::tab_header(title = "BS — Coordination and Demand choice 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_coord_gender_bs| BS — Coordination and Demand choice 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 = 1.000 ns |
| Coordination Part 1 | Session gender | χ²(sim.): p = 0.610 ns |
| Mutual Demand (D) Part 2 | Session gender | χ²(sim.): p = 1.000 ns |
| Mutual Demand (D) Part 1 | Session gender | χ²(sim.): p = 0.621 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_bs| BS — Choice and signal distributions | ||||
| 95% Clopper-Pearson CI | ||||
| Choice / Signal | n | N | % | 95% CI |
|---|---|---|---|---|
| Part 1 | ||||
| T | 8 | 32 | 25.0% | [11.5%, 43.4%] |
| D | 24 | 32 | 75.0% | [56.6%, 88.5%] |
| Signal | ||||
| T | 11 | 32 | 34.4% | [18.6%, 53.2%] |
| D | 21 | 32 | 65.6% | [46.8%, 81.4%] |
| Part 2 | ||||
| T | 11 | 32 | 34.4% | [18.6%, 53.2%] |
| D | 21 | 32 | 65.6% | [46.8%, 81.4%] |
Proportions by phase
Show code
p_bs_dist
Within-subject shift: McNemar test (Part 1 vs Part 2)
Show code
tab_mcnemar_bs |>
gt::gt() |>
gt::tab_header(
title = "McNemar test — BS: choice1_D vs choice2_D",
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 — BS: choice1_D vs choice2_D | |||
| Paired within-individual | |||
| χ² | p-value | n | Note |
|---|---|---|---|
| 0.3080 | 0.5791 | 32 | OK |
Note
A significant McNemar result (p < 0.05) indicates that pre-play cheap talk systematically shifted individual Demand (D) choice rates. In BS, neither D nor T is a dominant strategy — best responses are symmetric. Cheap talk can be credible here: a player who signals D (Demand) is announcing their intention to claim the P1-preferred equilibrium. If the opponent signals T (Yield), the combination D/T (P1 Preferred) or T/D (P2 Preferred) allows efficient coordination. Signals can therefore serve as commitment devices, helping couples resolve the equilibrium-selection problem.
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_bs
Show code
p_choice2_by_oppsig_bs
Show code
p_choice2_infoset_bs
Show code
p_follow_opp_bs
Conditioning on gender and role
Show code
tab_cond_sig_bs |>
gt::gt() |>
gt::tab_header(title = "BS — 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)| BS — 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 = 1.000 ns |
| Part 1 = T | Role | χ²(sim.): p = 0.675 ns |
| Signal = T | Gender | χ²(sim.): p = 0.456 ns |
| Signal = T | Role | χ²(sim.): p = 1.000 ns |
| Part 2 = T | Gender | χ²(sim.): p = 1.000 ns |
| Part 2 = T | Role | χ²(sim.): p = 0.472 ns |
Show code
p_cond_sig_bs
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_bs |>
gt::gt() |>
gt::tab_header(title = "BS — 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)| BS — 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 | 32 | 62.5% | [43.7%, 78.9%] |
| Signal consistent with Part 1 (signal = choice1)2 | 21 | 32 | 65.6% | [46.8%, 81.4%] |
| Strategy changed Part 1 → Part 2 (choice1 ≠ choice2)3 | 13 | 32 | 40.6% | [23.7%, 59.4%] |
| 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_bs |>
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 | 32 | 62.5% | [43.7%, 78.9%] | 0.2153 |
Note
In Battle of the Sexes, a signal is honest if the player’s Part 2 action matches what they signalled. Honesty in this game reflects commitment. A player who signals “Demand” and plays “Demand” is using cheap talk to credibly commit and force the opponent to yield. A player who signals “Demand” but then plays “Yield” is effectively “chickening out” of their threat. The aggregate honesty rate primarily captures the credibility of these strategic demands.
Strategic transition heatmaps
Show code
p_sankey_bs
Conditioning on gender and role
Show code
tab_cond_honest_bs |>
gt::gt() |>
gt::tab_header(title = "BS — 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)| BS — 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 = 0.715 ns |
| Signal honest | Role | χ²(sim.): p = 0.720 ns |
| Strategy changed | Gender | χ²(sim.): p = 0.477 ns |
| Strategy changed | Role | χ²(sim.): p = 0.478 ns |
Show code
p_cond_honest_bs
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 coordination outcomes at the couple level.
Belief accuracy distribution
Show code
p_belief_bar_bs
Hypothesis tests: beliefs, cognitive ability & coordination
| BS — 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 | 0.169 | 0.354 | 32 |
| H2 | Individual | Players who made more quiz errors have less accurate beliefs | Quiz errors [log(1+x)] | Belief accuracy (0–2) | Spearman ρ | Negative | 0.099 | 0.591 | 32 |
| H3 | Couple | Couples where both players have perfect beliefs coordinate more in Part 2 | Coord. Part 2 (0/1) | Both perfect beliefs (0/1) | Fisher exact + φ1 | Positive | 0.424 | 0.213 | 16 |
| 1 H3 stat = phi coefficient (φ); p-value from two-sided Fisher exact test on 2×2 contingency table. | |||||||||
Note
H1 tests whether more reflective players (higher CRT) are better at predicting their opponent — 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 coordinate in Part 2 — the only couple-level hypothesis; uses Fisher exact given small N and binary outcomes. Note that H1 and H2 operate at the individual level while H3 is at the couple level; they are not directly comparable.
Conditioning on gender and role
Show code
p_cond_belief_bs
5 — Econometric models
5.1 — Determinants of belief accuracy
Estimate an ordered logit (proportional-odds model) for the belief accuracy score (0 = both wrong, 1 = one correct, 2 = both correct) using Theory of Mind (MASC) and IRI subscales as predictors. The proportional-odds assumption implies a single log-odds shift per unit increase in each predictor, shared across both thresholds (0→1 and 1→2).
WarningSmall-sample caveat
With n = 32 participants (score 0: n=5; 1: n=16; 2: n=11), EPV is computed as min(n₀, n₂) / k: M1 = 5, M2 = 2.5, M3 = 1.7. All are well below the recommended 10. All results are exploratory and should be treated as hypothesis-generating.
Model specifications
\[ \begin{aligned} \text{M1:} \quad & \text{logit}\,P(Y \le j) = \alpha_j - \beta_1\,\text{MASC}_z \\ \text{M2:} \quad & \text{logit}\,P(Y \le j) = \alpha_j - \beta_1\,\text{MASC}_z - \beta_2\,\text{IRI-PT}_z \\ \text{M3:} \quad & \text{logit}\,P(Y \le j) = \alpha_j - \beta_1\,\text{MASC}_z - \beta_2\,\text{IRI-PT}_z - \beta_3\,\text{IRI-PD}_z \end{aligned} \]
MASC = Theory of Mind total score; IRI-PT = Perspective Taking subscale (cognitive empathy — most directly linked to predicting opponents’ decisions); IRI-PD = Personal Distress subscale (self-oriented distress — may impair strategic prediction). All scores z-standardised. OR > 1 shifts probability towards higher belief accuracy.
Coefficient table
Show code
tab_olog_gt| BS — Ordered logit: determinants of belief accuracy | |||||||
| DV = belief accuracy score (0/1/2, ordered). n = 32 (score 0: n=5; 1: n=16; 2: n=11). 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) | 0.316 | 0.349 | 0.904 | 0.3661 | 1.371 | 0.691 | 2.719 |
| M2: MASC + IRI-PT | |||||||
| MASC ToM score (z) | 0.269 | 0.359 | 0.751 | 0.4529 | 1.309 | 0.648 | 2.695 |
| IRI Perspective Taking (z) | -0.283 | 0.365 | -0.775 | 0.4385 | 0.754 | 0.361 | 1.544 |
| M3: MASC + IRI-PT + IRI-PD | |||||||
| MASC ToM score (z) | 0.266 | 0.360 | 0.740 | 0.4595 | 1.305 | 0.644 | 2.690 |
| IRI Perspective Taking (z) | -0.276 | 0.369 | -0.750 | 0.4531 | 0.758 | 0.361 | 1.562 |
| IRI Personal Distress (z) | 0.044 | 0.342 | 0.129 | 0.8977 | 1.045 | 0.527 | 2.050 |
| 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| BS — 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 | 32 | 1 | 5.0 | 69.4 | 0.0129 |
| M2: MASC + IRI-PT | 32 | 2 | 2.5 | 70.8 | 0.0223 |
| M3: MASC + IRI-PT + IRI-PD | 32 | 3 | 1.7 | 72.8 | 0.0225 |
Forest plot: odds ratios
Show code
p_forest_olog
5.2 — Demand choice when opponent signals Demand
Models
\[ \begin{aligned} \text{M1:} \quad & \text{logit}\,P(D) = \beta_0 + \beta_1\,\text{quiz\_err} \\ \text{M2:} \quad & \text{logit}\,P(D) = \beta_0 + \beta_1\,\text{quiz\_err} + \beta_2\,\text{CRT} \\ \text{M3:} \quad & \text{logit}\,P(D) = \beta_0 + \beta_1\,\text{CRT} \end{aligned} \]
Sample: opp_signal_received = T only. n = 21, events = 12. EPV: M1 = 12, M2 = 6, M3 = 12. All estimated with Firth penalised logit.
Results
| BS — Demand (D) when opponent signals T (Hare) | |||||||
| DV = choice2=D | opp_signal=D. n=21, events=12. EPV: M1=12, M2=6, M3=12. 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)] | -0.112 | 0.523 | -0.21 | 0.831 | 0.89 | 0.30 | 2.64 |
| M2: Quiz + CRT | |||||||
| Quiz errors [log(1+x)] | -0.133 | 0.537 | -0.25 | 0.804 | 0.88 | 0.31 | 2.51 |
| CRT score (0–4) | -0.203 | 0.483 | -0.42 | 0.674 | 0.82 | 0.32 | 2.10 |
| M3: CRT only | |||||||
| CRT score (0–4) | -0.250 | 0.477 | -0.52 | 0.600 | 0.78 | 0.28 | 1.95 |
| 1 Firth penalised logit used for all models (EPV < 10). OR > 1 → increases P(Demand (D) | opp signals D). 95% Wald CI. | |||||||
