Either prove or disprove each of the following statements in the context of 2 × 2 games (you may find it helpful to do proofs by providing examples or counter examples). 

a. If a player i has a dominant strategy in a game, then in every Nash equilibrium of that game player i will choose a dominant strategy. 

b. Every dominant strategy equilibrium of a game is a Nash equilibrium. 

c. Every Nash equilibrium of a game is a dominant strategy equilibrium. 

d. If a game outcome maximises utilitarian social welfare, then is Pareto efficient. 

e. If a game outcome is Pareto efficient, then it maximises utilitarian social welfare. 

f. If all utilities in a game are positive, then any outcome that maximises the product of utilities of players is Pareto efficient. 

g. If all utilities in a game are positive, then any Pareto efficient outcome of the game will maximise the product of utilities of players.