Abstract
In the last chapter, we saw that game theory is a powerful tool in dealing with the economic problems, especially when there are a small number of economic agents with conflicts of interest. Besides the issue of externalities, game theory is particularly useful for economic problems under imperfect and incomplete information.
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Notes
- 1.
We can actually allow infinite steps, infinite possible actions, and infinite players.
- 2.
In game theory, if Nature is involved in a game, Nature typically moves first. In particular, in a game of incomplete information, Nature always moves first.
- 3.
Kuhn’s Theorem has been extended in the literature to infinite games with infinite actions and horizon. However, perfect recall is always required for the equivalence result to hold.
- 4.
Condition (2) is usually enough to determine a mixed strategy NE. However, there are cases in which one player strictly prefers one strategy no matter what the other player does; in this case, (2) never happens. That is, one player plays a strategy with probability 1 in equilibrium. In this case, we need (3) to identify the equilibrium.
- 5.
We can also prove the necessity in the following way. If either (2) or (3) does not hold, then there are strategies \( s_{ki} ,s_{ji} \in {\mathbb{S}}_{i} \) with \( \sigma_{i} \left( {s_{ki} } \right) > 0 \) such that \( u_{i} \left( {s_{ki} ,\sigma_{ - i}^{*} } \right) < u_{i} \left( {s_{ji} ,\sigma_{ - i}^{*} } \right). \) If so, player \( {\text{i}} \) would do better by playing \( s_{ji} \) whenever he is supposed to play \( s_{ki} \) in the original NE, implying that the original equilibrium is not an equilibrium.
- 6.
Other players’ strategies may be functions of some random variables. In this case, these functions are required to be known to the player.
- 7.
This result does not hold when \( n \ge 3. \)
- 8.
Here, sequential rationality needs to be satisfied for all information sets, including those on off-equilibrium paths. This is important to know since, even with a proper belief system, some Nash equilibria do not satisfy sequential rationality on off-equilibrium paths. By Proposition 7.13, this happens when a NE is not a BE.
- 9.
The words “whenever applicable” in Gibbons’ (1992) definition are vague. A strategy is only a plan. Even for an equilibrium strategy, a planned action may never be taken in equilibrium. For example, given the equilibrium strategy profile \( \sigma^{*} \) defined by (12), for point \( y \) in Fig. 23, we have \( { \Pr }(y|H) = 1. \) But, \( \mu_{2}^{*} = 0. \) We can either say that \( \sigma^{*} \) is a BE since \( H \) is an off-equilibrium path or say that \( \sigma^{*} \) is not a BE since \( { \Pr }(y|H) \ne \mu_{2}^{*} . \) The weak version of BE in this book takes the former definition, while the strong version of BE in Gibbons (1992) takes the latter definition.
- 10.
When we mention “consistency” without specifying which type of consistency, we refer to “equilibrium-path consistency”.
- 11.
We have expected payoffs in the cells since Nature offers a half-half chance. This NE is actually a BNE.
- 12.
See Mas-Colell et al. (1995, p. 285).
- 13.
Given that everyone else is staying on the equilibrium path, those players whose information sets are on off-equilibrium paths have no need to deviate from their equilibrium strategies (since what they do does not matter assuming that others will stick to their current strategies). Thus, for a NE, we only need to verify rationality for those information sets on the equilibrium path. In other words, if rationality is guaranteed on the equilibrium path, we have a NE. Since we do not solve for a NE by backward induction, there is no guarantee that a NE is rational on off-equilibrium paths. In Example 7.15, for NE \( \sigma^{*} = \left( {L,L} \right), \) the initial decision node is on the equilibrium path and P1 is indeed rational at that node. But the second decision node is on an off-equilibrium path and P2 is not rational. In fact, given P1 choosing \( {\text{L}}, \) P2 is indifferent among his choices. P2 simply reasons that since the equilibrium path will not pass through his information set, whatever he decides to do does not matter and he has no need to deviate from the equilibrium strategy. The situation in Example 7.17 for NE \( \sigma^{*} = \left( \left\langle L_{1} ,\hat{L}_{1}\right\rangle, L_{2} \right) \) is also the same.
- 14.
No idea what Gibbons actually means by “where possible”, at least our definition is consistent with the examples in Gibbons (1992, 180–183).
- 15.
A trembling-perfect BE is different from Selton’s (1975) trembling-perfect NE for an extensive-form game. NE does not involve beliefs.
- 16.
We can assume that Nature’s odds are always strictly positive. That is, if \( \rho = \left( {\rho_{1} , \ldots ,\rho_{k} } \right) \) is Nature’s probability distribution over an information set, we will have \( \rho_{i} > 0 \) for all \( i. \) There is no point for Nature to have zero probability.
- 17.
It is from Kreps and Wilson (1982).
- 18.
See the definition in Gibbons (1992, p. 236), called strict dominance. For games of incomplete information, each type of a player is treated as a separate player in the definition of complete dominance.
- 19.
More clearly, the “remaining part” means the remaining part relevant to the calculation of payoffs implied by the action at the node. Note that the same action can be taken at different nodes in the same information. Hence, it is an action and a node together that determine the remaining part of the game tree. Player \( i \) may be one of the subsequent players.
- 20.
This condition is Requirement 5 in Gibbons (1992, p. 235).
- 21.
Player \( i \) may take a mixed behavior strategy at this information set \( H \) in equilibrium. However, this action \( a_{i} \) will never be used by player \( i \) in equilibrium. If player \( i \) places a positive probability on \( a_{i} \) in equilibrium, as indicated by (2), this \( a_{i} \) will be indifferent to player \( i \)’s other actions used in equilibrium at \( H \). This means that player \( i \) will do equally well by placing 100% probability on this action, given the remaining part of \( \sigma^{*} \). This is impossible by condition (16).
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Wang, S. (2018). Imperfect Information Games. In: Microeconomic Theory. Springer Texts in Business and Economics. Springer, Singapore. https://doi.org/10.1007/978-981-13-0041-7_7
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