Final answer:
The maximum sum of players' payoffs in the first period depends on the specific payoff matrix, but rational players would negotiate for a sum that optimizes the proposer's payoff while being acceptable to the responder.
Step-by-step explanation:
The student's question appears to be related to game theory in economics, specifically focusing on the outcomes of strategies in the first period of a game involving payoffs between players. If we consider the situation where both players are rational and have knowledge of the payoffs, the likely outcome in the first period is that a player will make an offer that maximizes their own payoff while still being acceptable to the other player to ensure cooperation and prevent rejection. In reality, however, experimental data has shown that players tend to deviate from purely self-interested behavior and often seek fairness, resulting in offers that are more equitable, such as a $7-$3 split, rather than the minimum amount they might get away with offering.
In summary, the maximum sum of the players' payoffs that can be achieved in the first period will depend on the specific payoff matrix or game structure, but rational players would likely negotiate to achieve a sum that reflects the highest payoff for the proposer that the responder is willing to accept. This is under the assumption of complete information and rational behavior.