I wonder whether this one is true (and can be easily proved): For a normal form game G and actions ai for a player i, removing a set of actions a−i from the game yields a game G− in which the Nash equilibria are worse on average for i (or alternatively the pareto-best/pareto-worst Nash equilibrium is worse for G− than for G).
It’s false: consider the normal form game
(0,0) (2,1)
(1,1) (3,0)
For the first player the first option is dominated by the second, but once the second player knows the first player is going to choose the second option, he’s motivated to take the first option. Removing the first player’s second option means the second player is motivated to take the second option, yielding a higher payoff for the first player.
It’s false: consider the normal form game
(0,0) (2,1)
(1,1) (3,0)
For the first player the first option is dominated by the second, but once the second player knows the first player is going to choose the second option, he’s motivated to take the first option. Removing the first player’s second option means the second player is motivated to take the second option, yielding a higher payoff for the first player.