The game works as follows: two player are involved in a dispute over an item. the value of the object to player i is vi>0. time is modeled as a continuous variable that starts at 0 and runs indefinitely. Each player chooses when to concede the object to the other player; if the first player to concede does so at time t,the other player obtains the object at that time.If both players concede simultaneously, the object is split equally between them, player i receives a payoff of vi/2. until the first concession each player loses one unit of payoff per unit of time.
So , I have made the game into strategic game. I still have a slight suspicion that it is wrong for the (-t,-t).
(v/2-t)(-t,v-t)
(v-t,-t)(-t,-t)
where the first action for both player are concede and the second action is wait .
How do I find pareto optimal payoff vectors from here?
From your definition of the game, it does not look like a lose-lose (-t,-t) is possible, so there are really only 3 entries in the table.
The easiest Pareto optimum is when both bid nothing and split the reward (v>0), anyone bidding will make at least one person worse off.
However, each player stands to gain if they bid the the greatest amount, and the loser bids less than v/2. As lose-lose is not an option, the only payoff to compare it to is the shared win. To move to the shared win from a winning position, the winning player will be worse off if the loser has bid less than v/2, so they also can be Pareto optima.