納什均衡與 subgame perfect equilibrium 的區別？

在什麼情況下納什均衡不是subgame perfect equilibrium

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在拓展式博弈(extensive form)中，NE和SPE的區別在於威脅是否可信。

比如最經典的行業進入博弈：

B選擇不進入(2,1)

B選擇進入

- A選擇競爭(0,0)

- A選擇默許(1,2)

NE有兩個：（默許，進入）和（競爭，不進入）

SPE只有一個：（默許，進入）

我們來觀察一下（競爭，不進入）這個NE而非SPE，其潛在的假設是，無論B選擇什麼，A一定選擇競爭（即NE所要求的單方面偏離），我們可以認為這是A在一開始提出的威脅，迫使B選擇不進入。

但是這個威脅是不可信的（在沒有其它附加條件的情況下），因為A在B之後進行選擇，如果B已經選擇了進入，A選擇默許比選擇競爭更優。

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Example:

Suppose there is an incumbent firm, I, and a potential entrant, E. The potential entrant first decides whether or not to enter the market and its entry or not entry is observed by

the incumbent.

In the last stage of this game the incumbent decides whether to fight entry (e.g. engage in an aggressive pricing strategy) or to accommodate entry.

Each firm therefore has two strategies:

? E: enter/stay out

? I : fight/accommodate if entry occurs

The payoffs are as follows:

? When E stays out: I= 2, E = 0.

? When E enters and I fights: I = ?1 and E = ?1.

? When E enters and I accommodates: I = 1 and E = 1.

Two pairs of strategies are Nash equilibria: {stay out, fight if entry} and {enter, acco. if entry}

One of these equlibria, namely {stay out, fight if entry} is somewhat strange.

It is clearly a NE for this game, but it is based on an empty (or non-credible) threat: the incumbent will never choose to fight once the potential entrant has entered the market.

Therefore, it is not very likely that the potential entrant will stay out of the market.

To rule out equilibria based on empty threats we need a stronger equilibrium concept for sequential games: subgame-perfect equilibrium.In this case,one of the Nash equilibriums is not subgame-perfect equilibrium.

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