= Solution
Use the standard <repeated game> convention that the first round's joint action is observed before round two. The following original <game tree> represents simultaneous moves by making row move first in each round while hiding that current move from column. Terminal labels are \b[total <payoffs> over both rounds], ordered row then column.
\Image[/past-exam-of-the-msc-in-advanced-computer-science-course-of-the-university-of-oxford/2021/computational-game-theory-two-round-tree.png]
{title=The full two-round extensive-form game, with all sixteen terminal payoff pairs and five column information sets}
Formally, round one starts at a row decision with actions $T,B$. The two ensuing column nodes form one <information set> $I_2^0$, with available actions $L,R$; column cannot tell which current row action occurred. The four completed first-round histories are $H=\{TL,TR,BL,BR\}$. At each $h\in H$, round two begins at a separate row node. Its two successors, after row's current $T$ or $B$, form a column <information set> $I_2^h$. Distinct $h$ have distinct <information sets>: both players know the whole first-round outcome. There are sixteen terminal histories $(h,k)\in H^2$, whose total <payoff> is $u(h)+u(k)$, as displayed in the tree.
The row player's five <information sets> are the initial singleton and the four singleton round-two nodes, one per $h$. The column player's five <information sets> are $I_2^0$ and the four two-node sets $I_2^h$. Thus each player must specify one binary choice at each of five <information sets>, including histories that its own first choice might prevent:
$$
\boxed{|S_1|=|S_2|=2^5=32.}
$$
Counting only the four possible two-round action sequences would miss the distinct contingent responses to the opponent's first action.
For a concrete pair of complete <pure strategies>, order the second-round histories as $(TL,TR,BL,BR)$ and use
$$
s_1=(T;B,T,T,T),\qquad s_2=(L;R,L,L,L).
$$
The first entry is the initial action, and the four remaining entries prescribe the action after each first-round history. This induces the unique path $TL$ in round one followed by $BR$ in round two. Hence the total <payoff> is
$$
\boxed{(1,5)+(2,1)=(3,6).}
$$
The off-path entries are part of each <pure strategy>, even though this pair does not reach them.
For randomization, a <mixed strategy> is a <probability distribution> over the $32$ complete plans, with $31$ free <probabilities> in a general distribution. A <behavioural strategy> instead gives one binary-action <probability> at each of five <information sets>, requiring five parameters. This game has <perfect recall>: a player never forgets a previous own action or a first-round observation, and the only concealed action is the opponent's current simultaneous choice. Consequently \b[neither kind of randomization has a greater attainable-payoff advantage]: <Kuhn's theorem> gives <realization-equivalent strategies> against every fixed opposing strategy.<Extensive games and the problem of information by Harold W. Kuhn>[original article, pp. 193–216, equivalence under perfect recall]{ref}
The equivalence can be verified directly here. Given a <behavioural strategy>, sample all five contingent actions independently before play; the resulting product <probability distribution> over complete plans is an equivalent <mixed strategy>. Conversely, let $\mu$ be a <mixed strategy> for player $i$. Use its marginal <probability> for the initial action. At history $h$ containing own first action $a_i$, choose second action $b_i$ with <conditional probability>
$$
\beta_i(b_i\mid h)=
\frac{\mu\{s:s^0=a_i,\ s^h=b_i\}}
{\mu\{s:s^0=a_i\}},
$$
whenever the denominator is positive; if it is zero, choose arbitrarily at that unreachable history. The opponent's first-round action is simultaneous and cannot depend on the sampled hidden plan beyond the chosen first action. Thus conditioning on $h$ preserves exactly these <conditional probabilities> of actions, and multiplying the first- and second-round factors reproduces each terminal <probability>. Correlations between prescriptions at mutually exclusive second-round histories have no further observable effect. <Behavioural strategies> offer the more compact representation, while <mixed strategies> describe a single random choice of a full plan. Imperfect information here does not mean imperfect recall.
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