4. Supposing there are three players, each player is given a bag and asked to contribute in his own money with one of the three amount {$0, $3, $6}. A referee collects all the money from the three bags and then doubles the amount...
4. Supposing there are three players, each player is given a bag and asked to contribute in his own money with one of the three amount {$0, $3, $6}. A referee collects all the money from the three bags and then doubles the amount using additional money. Finally, each player share the whole money equally. For example, if both Players 1 and 2 put $0 and Player 3 puts $3, then the referee adds another $3 so that the total becomes $6. After that, each player will obtain $2 at the end. Every player want to maximise his profit, but he does not know the amount contributed from other players. [Hint: profit = money he obtained - money he contributed.]
(a) Compute the profits of each player under all strategy combinations and make the payoff matrix for the three players. [Hint: you can create multiple payoff tables to demonstrate the strategy combinations. The referee is not a player and should not be in the payoff table.]
[5 Marks]
(b) Find the Nash equilibrium of this game. What are the profits at this equilibrium? Explain your reason clearly.
[5 Marks]
(c) Whether the Nash equilibrium of this game provides the best strategy (Pareto op timal solution) for all players if they can cooperate? Explain the relationship between Nash Equilibria and Pareto optimality in 3-5 sentences.
[5 Marks]
(d) Describe a real life example in which the cooperation should be the name of the game. [5 Marks]
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