diff --git a/src/game.rs b/src/game.rs
index d943846..1b9ede3 100644
--- a/src/game.rs
+++ b/src/game.rs
@@ -1322,4 +1322,99 @@ mod tests {
         assert_ulps_eq!(p[1][0], post_2vs1[1][0], epsilon = 1e-6);
         assert_ulps_eq!(p[1][1], t_b[1].prior, epsilon = 1e-6);
     }
+
+    #[test]
+    fn run_chain_honours_max_iter_in_convergence_options() {
+        let players: Vec<R> = (0..4).map(|_| R::default()).collect();
+        let teams: Vec<Vec<_>> = players.iter().map(|p| vec![*p]).collect();
+        let result = vec![3.0, 2.0, 1.0, 0.0];
+        let weights = vec![vec![1.0]; 4];
+
+        // Capped at 1 iteration: cannot fully propagate down a 4-team chain.
+        let mut arena = ScratchArena::new();
+        let g_capped = Game::ranked_with_arena(
+            teams.clone(),
+            &result,
+            &weights,
+            0.0,
+            crate::ConvergenceOptions {
+                max_iter: 1,
+                ..crate::ConvergenceOptions::default()
+            },
+            &mut arena,
+        );
+        let posteriors_capped = g_capped.posteriors();
+
+        // Same inputs, plenty of iterations: fully converged.
+        let mut arena = ScratchArena::new();
+        let g_full = Game::ranked_with_arena(
+            teams,
+            &result,
+            &weights,
+            0.0,
+            crate::ConvergenceOptions::default(),
+            &mut arena,
+        );
+        let posteriors_full = g_full.posteriors();
+
+        // The two posteriors should differ — capped did not converge.
+        let mut max_diff: f64 = 0.0;
+        for (team_capped, team_full) in posteriors_capped.iter().zip(posteriors_full.iter()) {
+            for (g_capped, g_full) in team_capped.iter().zip(team_full.iter()) {
+                max_diff = max_diff.max((g_capped.mu() - g_full.mu()).abs());
+                max_diff = max_diff.max((g_capped.sigma() - g_full.sigma()).abs());
+            }
+        }
+        assert!(
+            max_diff > 1e-6,
+            "max_iter=1 should differ from full convergence; max_diff={max_diff}"
+        );
+    }
+
+    #[test]
+    fn run_chain_with_damping_converges_to_same_posterior() {
+        let players: Vec<R> = (0..4).map(|_| R::default()).collect();
+        let teams: Vec<Vec<_>> = players.iter().map(|p| vec![*p]).collect();
+        let result = vec![3.0, 2.0, 1.0, 0.0];
+        let weights = vec![vec![1.0]; 4];
+
+        let mut arena = ScratchArena::new();
+        let g_undamped = Game::ranked_with_arena(
+            teams.clone(),
+            &result,
+            &weights,
+            0.0,
+            crate::ConvergenceOptions::default(),
+            &mut arena,
+        );
+        let posteriors_undamped = g_undamped.posteriors();
+
+        // alpha=0.5 with extra iterations: should reach the same fixed point.
+        let mut arena = ScratchArena::new();
+        let g_damped = Game::ranked_with_arena(
+            teams,
+            &result,
+            &weights,
+            0.0,
+            crate::ConvergenceOptions {
+                alpha: 0.5,
+                max_iter: 100,
+                ..crate::ConvergenceOptions::default()
+            },
+            &mut arena,
+        );
+        let posteriors_damped = g_damped.posteriors();
+
+        let mut max_diff: f64 = 0.0;
+        for (team_u, team_d) in posteriors_undamped.iter().zip(posteriors_damped.iter()) {
+            for (g_u, g_d) in team_u.iter().zip(team_d.iter()) {
+                max_diff = max_diff.max((g_u.mu() - g_d.mu()).abs());
+                max_diff = max_diff.max((g_u.sigma() - g_d.sigma()).abs());
+            }
+        }
+        assert!(
+            max_diff < 1e-4,
+            "α=0.5 should reach the same fixed point as α=1.0; max_diff={max_diff}"
+        );
+    }
 }