1. ECO290E: Game Theory Lecture 10 Examples of Dynamic Games

2. Review of Lec. 8 & 9 A strategy in a dynamic game is a complete plan of actions, i.e., specifying what she will do in every her information set. Dynamic games may have many Nash equilibria, but some of these may involve non-credible threats or promises. The subgame perfect Nash equilibria are those that pass a credibility test.

3. SPNE A subgame perfect Nash equilibrium (SPNE) is a combination of strategies in a extensive-form which constitutes a Nash equilibrium in every subgame.  Since the entire game itself is a subgame, it is obvious that a SPNE is a NE, i.e., SPNE is stronger solution concept than NE.

4. Existence of NE Since any dynamic game has a (unique) normal-form representation, at least one NE exist as long as a game is finite, i.e., finite number of players and strategies. If some of the paths in the game tree involve infinite stages (information set), then the corresponding normal-form game is no longer finite.  The dynamic game may not have a NE!

5. Existence of SPNE Any finite dynamic game also has a SPNE, since each subgame is finite and thereby must have at least one NE. To find a SPNE, first identify all the smallest subgames that contain the terminal nodes, and then replace each such subgame with the payoffs from one of Nash equilibria. The next step is to solve the second smallest subgames given these truncated payoffs. Working backwards through the tree in this way yields a SPNE!

6. Other Examples To be completed.