Please consider the following exercise:

A coin with probability $2/3$ of heads and $1/3$ of tails is tossed and the outcome is shown to player I but not to player II. Player I then makes a claim which may be true or false that the coin turned up heads or that the coin turned up tails. Then, player II, hearing the claim, must guess whether the coin came up heads or tails. Player II wins $\$3$ if his guess is correct, and nothing otherwise. Player I wins $\$3$ if I has told the truth in his claim. In addition, player I wins an additional $\$6$ if player II guesses heads. Draw the Kuhn tree of this game and put the game into strategic (bimatrix) form.

The solution: The Kuhn tree and the corresponding game matrix should look like this:

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When I was trying to solve this exercise I managed to get the Kuhn tree right, but the game matrix confuses me. The rows of the matrix should correspond to player I's strategies and the columns should correspond to player II's strategies. The first row corresponds to player I's strategy to say 'Heads' when the coin was actually 'Heads' and the second row corresponds to player I's strategy of saying 'Tails' when the outcome of the coin actually was 'Heads'. Each element in the matrix represents the respective payoff to both players. The first element corresponds to the payoff when player I chooses HH and player II chooses HH; in that case player I gets $8$ and player II gets $2$.

My question: I have some trouble understanding some of the outcomes in the game matrix. The first element makes sense to me: there's a probability of $2/3$ that the coins comes up heads, so the expected payoff when player I says heads when the coin is heads and player II says heads when player I says heads is $2/3 * (9,3)$. However, I have no idea what to make of the element in row $2$ and column $1$, $(9,2)$. How do you derive this value? Since there's only two options that pay $9$ to player I, this $(9,2)$ is derived by doing $2/3 * (9,3) + 1/3 *(9,0) = (9,2)$. But why should this make sense? How does this correspond to the situation where player I plays HT and player II plays HH?


Row 2, column 1 of the matrix gives the expected payoff when player I plays strategy $HT$ (i.e. claim $H$ if the coin comes up $H$, and $T$ if the coin comes up $T$) and player II plays the strategy $HH$ (i.e. guess $H$ no matter I's claim).

Player I's (certain) payoff given II's strategy is 9 because the outcomes $HHH$ (nature chooses $H$, I claims $H$, II guesses $H$) and $TTH$ (nature chooses $T$, I claims $T$, II guesses $H$) both give a payoff of 9 (because player I wins 3 for telling the truth and an additional 6 because player II guesses $H$).

Player II's expected payoff given I's strategy is 2 because, given player I is telling the truth (his claims match nature's choice) is $$\frac{2}{3}3+\frac{1}{3}0=2$$ (The payoff $3$ comes from the first terminal node. The payoff $0$ comes from the fifth terminal node.)

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