Consider a research funding game in which two government agencies, the U.S. Department of Energy (DoE) and the Defense Advanced Research Projects Agency (DARPA), each decide which of two research projects ("batteries" or "solar") to fund. DoE has a dominant strategy, investing in batteries, but wants most for DARPA to invest in solar. DARPA prefers to invest in the same project as DoE, but wants most for DoE to invest in batteries. The ordinal payoff matrix for this game is shown below.

Batteries Solar DoE Batteries 2,4 4,3

Solar 1,1 3,2

1)Identify all pure-strategy Nash equilibria of this game (or explain why no such equilibrium exists) in the case when DoE and DARPA move simultaneously. 

2)Draw the game tree for the case when DoE moves first. What are the rollback equilibrium strategies and outcome? 

3)In the case when DoE moves first, is it possible for DARPA to achieve its best possible outcome by credibly declaring a strategic move in the pregame? If not, why not? And if so, what sort of strategic move would DARPA use, and how might it phrase its declaration? 

4)Draw the game tree for the case when DARPA moves first. What are the rollback equilibrium strategies and outcome? 

5) In the case when DARPA moves first, is it possible for DoE to achieve its best possible outcome by credibly declaring a strategic move in the pregame? If not, why not? And if so, what sort of strategic move would DoE use, and how might it phrase its declaration?