Help with 2x2 strategic game with 1 player

Thanks for all your help and patience! I used a different method to solve it and arrived at a different conclusion. I looked at the outcome of this game if it was played 100 times and calculated the expected payoff of the signal that way.

Out of the 100 times, the signal will be true 75 of the times and untrue 25 of the times. Out of the 75 times that the signal is true, the actual outcome of the first game (in regards to the party leader's opinion being correct or not) will be correct 37 times and incorrect 38 times (50% probability for each). Having calculated the expected payoffs considering the search costs, the payoffs are now -1 for both Challenge and Don't challenge. In the 37 times that the signal is true and that it predicts that the leader's opinion is correct, the member will choose not to challenge and receive a payoff of -1.In the 38 times when the signal is true and predicts that the leader's opinion is wrong, the member will Challenge and receive a payoff of 3.

Out of the 25 times that the signal is false, 13 of those times the party leader's opinion will be correct and 12 of those times it will be incorrect. In the 13 times that the leader is correct, and the member thinks the leader will be incorrect (since the signal says so, but is actually false) she will choose to challenge and will receive a payoff of -5. In the 12 times that the leader is incorrect, and the member wrongly thinks the leader will be correct, she will not challenge and receive a payoff of -1.

Expected payoffs: ((37 * -1) + (38 * 3)) / 75 = 0.52 ((13 * -5) + (12 * -1)) / 25 = -2.56

Weighted average: ((75 * 0.52) + (25 * -2.56)) /100 = -0.25

So the member will be worse off choosing to search further (using the signal) than when she just plays the game without the signal. Or have I gone wrong somewhere in my reasoning? Thanks again for all your help!

/r/GAMETHEORY Thread Parent