4 ms·

What happens if Bob lies to Alice 20% of the time and Alice lies to me 20% of the time but I only get input from Alice?

• kmm 9mo ago
  In that case following Alice's input is still the best strategy, but you'll be worse off: you'd only be right if both tell the truth, at 80%80%=64%, or both lie, at 20%20%=4%, for a total of 68%. In the general case of n intermediate occasional liars, the odds of the final result being accurate goes to 50% as n grows large, which makes sense, as it will have no correlation anymore to the initial input.
  • saila 9mo ago
    Thanks. I came up with this Python simulation that matches your 68%: import random def lying_flippers(num_flips=1_000_000): """ - Bob flips a coin and tells Alice the result but lies 20% of the time. - Alice tells me Bob's result but also lies 20% of the time. - If I trust Bob, I know I'll be correct 80% of the time. - If I trust Alice, how often will I be correct (assuming I don't know Bob's result)? """ # Invert flip 20% of the time. def maybe_flip_flip(flip: bool): if random.random() < 0.2: return not flip return flip def sum_correct(actual, altered): return sum(1 if a == b else 0 for (b, a) in zip(actual, altered)) half_num_flips = num_flips // 2 twenty_percent = int(num_flips * 0.2) actual_flips = [random.choice((True, False)) for _ in range(num_flips)] num_heads = sum(actual_flips) num_tails = num_flips - num_heads print(f"Heads = {num_heads} Tails = {num_tails}") bob_flips = [maybe_flip_flip(flip) for flip in actual_flips] alice_flips = [maybe_flip_flip(flip) for flip in bob_flips] bob_num_correct = sum_correct(actual_flips, bob_flips) bob_percent_correct = bob_num_correct / num_flips alice_num_correct = sum_correct(actual_flips, alice_flips) alice_percent_correct = alice_num_correct / num_flips # Trusting Bob should lead to being correct ~80% of the time. # This is just a verification of the model since we already know the answer. print(f"Trust Bob -> {bob_percent_correct:.1%}") # Trusting Alice should lead to being correct ?% of the time. # This model produces 68%. print(f"Trust Alice -> {alice_percent_correct:.1%}") print()