Trembling hand perfect equilibrium (THPE)
Trembling hand perfect equilibrium (THPE) rules out Nash equilibriums where the strategies
become suboptimal when the opponents can “tremble” and accidentally play one of their other
strategies. More precisely, the THPE concept requires the Nash equilibrium to remain a Nash
equilibrium for a least one possible sequence of trembles for each of the opposing players. The
trembles must be “severe” enough that the opponents put at least a small probability on each of
their strategies and these probabilities only converge to zero in the limit.
Formally, assume that there are I players who use the strategies   1 ,  2 ,...,  I  . Some or all
of the strategies may be mixed strategies. Define a sequence of completely mixed strategies
indexed by n,  n  1n , 2n ,..., In  ; where “completely mixed” means that each player assigns a
positive probability to each of her pure strategies. Then if
(1) lim n    lim 1n , 2n ,...,  In   1,  2 ,...,  I 
n
n
(2)  i is a best response to  n,  1n , 2 n ,... i 1,n , i 1,n ..., In 
then   1 ,  2 ,...,  I  is a Trembling Hand Perfect Equilibrium.
Each player’s  i must therefore be a best response, both when the opponents tremble and play
the perturbed strategies and in the limit when they stop trembling and play their  strategies.
The requirement that a Nash equilibrium must be able to survive certain “trembles” by the other
players may be reasonable. Just like subgame perfect Nash equilibrium (SPNE) eliminates
“implausible” Nash equilibriums that are based on non-credible threats and promises, THPE
eliminates “implausible” Nash equilibriums that are not robust to mistakes in the strategy
choices.
Result 1: In games with only two only players, the trembling–hand restriction rules out Nash
equilibria based on a weakly dominated strategy. The text below provides an example.
Result 2: In games with more than two players, the trembling-hand restriction can rule out other
Nash-equilibriums as well, including ones without dominated strategies. The text below provides
an example.
Result 3: The THPE restriction does not necessarily help much in games with multiple
subgames as opposed to one-shot simultaneously move games. It may be better to first impose
subgame perfection (SPNE instead of NE) and only then impose robustness to trembles in every
subgame or at every information set (also called extensive-form THPE as opposed to normalform THPE). The text below provides an example.
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Example of Result 1: In games with only two only players, the trembling–hand restriction
rules out Nash equilibria based on a weakly dominated strategy..
R
L
P1
R
1,1 NE1
1,0
L
0,0
1,2 NE2
P2
The game above has two Nash equlibriums: {R,R} and {L,L}. However, since (0,1)  (1,1) , L is
weakly dominated for player 1. In the following, we will first show that NE1 is a THPE. Then
we show that NE2 is not a THPE.
*In order to show that NE1 is a THPE, suppose that the players initially play {R,R}. With
probability  player 1 trembles and plays L. With probability  , player 2 trembles and plays L.
Formally we can define the strategies
{ 1 ,  2 }  {R, R}
and the perturbed strategies
{ 1n , 2n }  { pr ( R)  (1   n ), pr ( R)  (1   n )} ,
where  n n1 and  n n1 are two convergent sequences with lim  n  lim  n  0 .


n
n
Player 1’s strategy is trembling-hand perfect if
Eu1 ( R)  (1   n )1   n 1  (1   n )0   n 1  Eu1 ( L)  0   n , which always holds.
Player 2’s strategy is trembling-hand perfect if
Eu 2 ( R)  (1   n )1   n 0  (1   n )0   n 2  Eu1 ( L)   n  1 / 3 .
We can ensure the latter condition with the probability sequence  n n1  1 / 3nn1 .


2
*In order to show that NE2 is NOT a THPE, suppose that the players initially play {L,L}.
Redefine  to be the probability that player 1 trembles and plays R. Redefine  to be the
probability that player 2 trembles and plays R. Formally we can define the strategies
{ 1 ,  2 }  {L, L}
and the perturbed strategies
{ 1n , 2n }  { pr ( L)  (1   n ), pr ( L)  (1   n )} ,
where  n n1 and  n n1 are again two convergent sequences with lim  n  lim  n  0 .


n
n
If we wanted to prove that NE2 was a THPE - rather than disprove it - we would just need it to
hold for particular sequences, such as  n n1   n n1  0.01 / nn1 . In order to disprove a



THPE, though, we have to show that no possible sequences exist. Nonetheless, we know that for
player 1’s expected payoff to L to exceed the expected payoff to R, we need
Eu1 ( L)  (1   n )1   n 0  (1   n )1   n 1  Eu1 ( R)  0   n ,
which is impossible when player 2’s tremble probability  n is strictly positive. Since player 1’s
L strategy in NE2 is not trembling-hand perfect, {L,L} is not a THPE.
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Example of Result 2: In games with more than two players, the trembling-hand restriction
can rule out other Nash-equilibriums as well, including ones without dominated strategies.
Assume that players 1 and 2 choose the row and column strategies as always. Player 3 chooses
the “box”.
PLAYER 3 L
P2 R
P1
R
L
PLAYER 3 R
L
R
L
1,1,1 1,0,0
P1
R
0,0,0
1,1,1
0,0,0
L
1,1,1
0,0,0
P2
1,1,0
In this game, player 1 lacks a dominated strategy since the feasible payoffs from R do not
dominate the payoffs from L, 1,1,0,1   0,1,1,0  , or vice versa. For player 2,
similarly, 1,0,0,1   0,1,1,0  and vice versa. For player 3, similarly, 1,0,0,0    0,1,1,0  .The
strategy combination {L,L,L} is therefore a Nash equilibrium without a dominated strategy.
In the following, we will show that the THPE concept eliminates the {L,L,L} equilibrium even
though it does not involve a dominated strategy. In order to show this, we will assume that player
1 trembles and plays R with probability  ; player 2 trembles and plays R with probability  ; and
player 3 trembles and plays R with probability  . In order for player 1’s L strategy to remain
optimal, the expected payoff to L must continue to exceed the expected payoff to R,
(1   )(1   )1   (1   )0   (1   )0  1  (1   )(1   )1   (1   )1   (1   )1   0
(1   )(1   ) 
  (1   ) 


  (1   ) 


 

T
0
 1
   0  3      0
 1
 
1 
If the tremble probabilities are precisely zero,     0 , the inequality of course holds, since
{L,L,L] with certainty is a Nash equilibrium. But if the tremble probabilities are not zero, the
1  1
inequality requires   
 . The problem with this is that the THPE concept requires the
3 3 3
L strategy to remain optimal as the opponents’ tremble probabilities converge to zero; and not
just when player 3’s tremble probability exceeds 1/3+  / 3 . It follows that {L,L,L} is not a
THPE. Intuitively, player 1’s L strategy is a non- dominated strategy because it is better than R
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when both of the other players deviate, so in the LRR scenario. But the probability of this
scenario is only  , which becomes small – and smaller than each of the unilateral tremble
probabilities  (1   ) and  (1   ) - as  and  decrease.
Example of Result 3: The THPE restriction does not necessarily help much in games with
multiple subgames as opposed to one-shot simultaneously move games. It may be better to
first impose subgame perfection (SPNE instead of NE) and only then impose robustness to
trembles in every subgame or at every information set (also called extensive-form THPE as
opposed to normal-form THPE).
Consider the following market entry game from the lecture notes of Albert Banal-Estanol
(http://albertbanalestanol.com/wp-content/uploads/gtcm-Lecture-2b.pdf). First the entrant firm
chooses to enter and then the entrant and the incumbent firm simultaneously decide whether to
accommodate or fight each other.
Accommodate if In Fight if In (F)
(A)
Incumbent
Entrant
Out, Accommodate if In (OA)
2,2
2,2 NE2
Out, Fight if In (OF)
2,2
2,2 NE3
In, Accommodate if In (IA)
4,1 NE1+SPNE
1,0
In, Fight if In (IF)
0,0
0,1
Entrant
Out
In
2,2
I
A
F
E
A
4,1
1,0
F
0,0
0,1
Backward induction shows that the only Nash equilibrium after “In” is {A,A}. Knowing this, the
entrant should choose “In”. Thus the only SPNE is {IA, A}. However, the normal form shows
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that NE2 and NE3, {OA, F} and {OF, F} are also Nash equilibriums. Moreover, the normal form
shows that none of the three NE’s use a dominated strategy. We can therefore construct trembles
that support NE2 and NE3 as THPEs. For example, to show that NE2 is a THPE we can assume
they play {OA,F} and choose tremble probabilities 1 ,  2 ,  3 for the entrant 1 and  for the
incumbent. In order for the incumbent’s strategy to remain optimal with the entrant’s trembles
we need
(1  1   2   3 )2  1 2   2 0   31  (1  1   2   3 )2  1 2   21   3 0   3   2
which we can solve with the probability series  3n n1  0.1/ nn1 ,  2 n n1  0.01/ nn1 .




In order for the entrant’s strategy to remain optimal we need
(1   )2   2  max (1   )2   2, (1   )1   4, (1   )0   0   2  max{2,1  3}  2
if we only limit the incumbent’s tremble probability to   1/ 3, for example with the sequence
n n1  1/ 3nn1 . Thus, NE2 is a THPE. We can similarly show that NE3 is a THPE. Thus,


the THPE concept does not eliminate any of the three Nash equilibriums, including the two
implausible ones NE2 and NE3, which are not subgame perfect.
In order to make the THPE concept more useful in this context with multiple subgames, we can
first impose subgame perfection and then look for THPE in each of the resulting subgames we
isolate. This way we say that we look for extensive-form THPE rather than normal form THPE.
In this example, the only Nash equilibrium after “In” is {AA}, which you can show is robust to
trembles. Intuitively, for the entrant, “A” is a dominant strategy, so she chooses it for any and all
Incumbent trembles. On the other hand, the incumbent prefers “A” as long as we make the
entrant’s tremble probability below 0.5. Since AA is the only THPE after entry, the extensiveform THPE concept means they have to play that after entry. In turn, the entrant chooses “In”.
we can conclude that {IA, A} is a unique extensive form THPE. It also coincides with the unique
SPNE in the game, which is nice.
(References/sources
http://dept.econ.yorku.ca/~sam/6100/slides/8c.pdf
http://albertbanalestanol.com/wp-content/uploads/gtcm-Lecture-2b.pdf
Kreps, David M., and Robert Wilson. "Sequential equilibria." Econometrica: Journal of the
Econometric Society (1982): 863-894.
Selten, Reinhard. "Reexamination of the perfectness concept for equilibrium points in extensive
games." International journal of game theory 4.1 (1975): 25-55).
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