Is it better to give information, receive it, or be ignorant in a two-player game?

The standard approach in a biological two-player game is to assume both players choose their actions independently of one another, having no information about their opponent's action (simultaneous game). However, this approach is not realistic in some circumstances. In many cases, one player chooses his action first and then the second player chooses her action with information about his action (Stackelberg game). We compare these two games, which can be mathematically analyzed into two types, depending on the direction of the best response function (BRF) at the evolutionarily stable strategy in the simultaneous game (ESSsim). We subcategorize each type of game into two cases, depending on the change in payoff to one player, when both players are at the ESSsim, and the other player increases his action. Our results show that in cases where the BRF is decreasing at the ESSsim, the first player in the Stackelberg game receives the highest payoff, followed by both players in the simultaneous game, followed by the second player in the Stackelberg game. In these cases, it is best to be the first Stackelberg player. In cases where the BRF is increasing at the ESSsim, both Stackelberg players receive a higher payoff than players in a simultaneous game. In these cases, it is better for both players to play a Stackelberg game rather than a simultaneous game. However, in some cases the first Stackelberg player receives a higher payoff than the second Stackelberg player, and in some cases the opposite is true.