 Solution

A security level payoff maximizes what a player guarantees against the opponent. Let $p$ be the row player's probability of the first action. Its guaranteed payoff isThe decreasing and increasing terms cross at $p=1/5$, attaining $d_1=12/5$; moving either way lowers the smaller term. For the column player, any mixture has zero payoff against the first row, while its second-row payoff is nonnegative. Therefore $d_2=0$ andThe feasible set is the convex hull of the four joint pure-action payoff vectors. Its upper Pareto frontier connects $(4,0)$ to $(3,2)$ to $(2,3)$. Write the row payoff as $u$ and column payoff as $v$. On the first Pareto frontier segment, $v=8-2u$ for $3\le u\le 4$. The Nash product isa concave quadratic with derivative $64/5-4u$, maximized at $u=16/5$, $v=8/5$, giving product $32/25$. On the portion of the other Pareto frontier segment satisfying bargaining individual rationality, $12/5\le u\le 3$ and $v=5-u$. Its product derivative $37/5-2u$ is positive throughout, so its largest product is $6/5$ at $(3,2)$, smaller than $32/25$. All dominated points can be discarded by Pareto efficiency. ConsequentlyThis payoff is implemented by a correlated payoff lottery choosing the payoff $(4,0)$ with probability $1/5$ and $(3,2)$ with probability $4/5$. The convex hull permits such lotteries over joint outcomes; it is not restricted to independent mixed strategies.