Discussion Paper No. 378

Commitment games revisited

Ryosuke ISHII

Nov. 2022

INSTITUTE OF ECONOMIC RESEARCH
Chuo UniversityTokyo, Japan

Commitment games revisited
Ryosuke ISHIIy
November 30, 2022

Abstract
Commitments have positive-negative e¤ects on games’outcomes, which vary depending on the conformation of the commitments. Renou (2009) investigates commitment in …nite strategic form games mainly to the extent of pure strategies, and themajor …nding can be applied to mixed strategies. Here, there is little or no randomizing commitment. We reconsider commitment games that have a one-shot commitment
before an action stage. There are two examples. The …rst has a mixed strategy equilibrium with players’randomizing in the commitment stage, which Pareto-dominates
all the pure commitment equilibria. The other has no pure commitment equilibrium.
In addition, we observe that some of the supports in a mixed strategy equilibriummight be the pure strategy equilibrium outcome after introducing multiple commitment stages.
JEL classi…cation: C72, C73
Keywords: Commitment, Existence, Mixed strategy equilibrium, Pareto dominance.

1

Review of Renou’s Commitment Setting

Renou (2009) has derived various implications of commitment games. The analysis isnovel since it considers committing not to a single action but to a set of actions. In thecommitment stage (…rst stage), players simultaneously commit to subsets of their action
sets, followed by the action stage (second stage) in which they choose actions among therestricted action sets. Renou (2009) investigates how introducing commitment in‡uences
outcomes in “status quo games”in a wide range of circumstances.
In this section, we verify commitment games as de…ned by Renou (2009) and the mainresults. Before playing a strategic form game (a status quo game,) players can makeI am grateful to Rohan Dutta for his comments and suggestions. Special thanks to Kazuhiko Hashimotofor his detailed comments on earlier drafts. I also thank seminar audiences at Chuo University. This work
was supported by JSPS KAKENHI Grant Numbers JP17K13704, JP20K01551, and Research Stimulus
Fund, Faculty of Economics, Teikyo University. Any remaining errors are my own responsibility.
yFaculty of Economics, Shimonoseki City University, 2-1-1, Daigaku-cho, Shimonoseki, Yamaguchi 7518510, Japan; +81(0)83-252-0288; ishii-ryo@shimonoseki-cu.ac.jp.

1commitments simultaneously. A commitment does not have to be a single strategy. Instead,
we may interpret commitment as the ability to eliminate some strategies in advance. A
player can do nothing (commit to all strategies in the status quo game), eliminate one
strategy (commit to all strategies but one), or keep a strategy and eliminate all others(commit to one strategy). A strategy pro…le of the status quo game made up by subgame
perfect strategies is called “implementable.”
Introducing a commitment device increases (or does not decrease, to be exact) purestrategy equilibria in all status quo games, since any pure Nash equilibrium is implementable in the status quo game. We con…rm this by examining a simple 2 2 status quogame.
aba 2; 2 0; 0b 3; 0 1; 1This status quo game is dominance solvable and has the unique Nash equilibrium (b; b).
For example, it is a subgame perfect equilibrium path that both players commit to fbg(eliminate a), followed by (b; b). There is no pro…table deviation for either player. Ifeither player leaves fa; bg, the opponent player must play b in the second stage, to which
the player with fa; bg chooses b and the outcome remains (b; b). Thus, this is a subgame
perfect equilibrium path. In most commitment games, any pure Nash equilibria in thecorresponding status quo games are implementable by all players committing to single
action sets that include only the Nash equilibrium strategies.
Conversely, (a; a), which is not a Nash equilibrium outcome in the status quo game isalso implementable. Consider the path below. Player 1 eliminates dominant strategy b
while player 2 does nothing in the …rst stage. In the second stage, player 1 must play a.
Player 2 compares payo¤ 2 with (a; a) and payo¤ 0 with (a; b), and chooses a, resultingin (a; a). A potential deviation for player 1 is doing nothing or committing to fbg in the…rst stage. However, the outcome of the deviation is (b; b) in each case, and thus it is not
a pro…table deviation. Coming back to the case of player 1 committing to fag, this path
constructs a subgame perfect equilibrium. Thus, (a; a) is implementable. In this manner,
the introduction of the commitment stage weakly increases pure strategy equilibria.

2Pareto Improvement through Randomized Commitments

In the last sentence on page 493, Renou (2009) states “although Theorem 2 is stated for·pure strategies, it also applies to mixed strategies”This
randomizing expansion requires
attention to some details, although what is unspeci…ed. We must de…ne other players’
randomized commitment X i as weighted “restricted status quo games by strategies ofother players, and player i best responds in the …rst stage toward the randomized others’commitments.”Notice that if we mistake X i as “all pure commitments”on the equilibrium
path, the latter claim of Theorem 2 does not hold.

2

Taking mixed strategies into account could Pareto-improve the outcome. Consider thenext example.
abcdeab9; 100; 110; 16; 50; 07; 010; 0 7; 110; 1 0; 1c1; 110; 03; 40; 14; 0GPd0; 00; 00; 52; 23; 0e1; 00; 61; 50; 01; 1This status quo game has the unique Nash equilibrium (e; e). The implementable strategy pro…les with pure commitments are (d; d) and (e; e) only. In the …rst stage, player 1
commits to fdg and player 2 does nothing, and (d; d) is realized.
There are no other implementable strategy pro…les. For example, (c; c) is not implementable. To make (c; c) a Nash equilibrium at an action stage, player 1 must eliminatee at the commitment stage, player 2, d and e. However, player 1 has an incentive to keep
_ By rights, player 2 has a punishing option ee, strictly improving her payo¤ from 3 to 4.
to avoid player 1’s deviation (leaving e.) However, player 2 has to eliminate e (and d) toelude her temptation to deviate to e (or d) from (c; c). A similar argument holds on (a; a).
Swapping the players explains that (b; b) is not implementable. Most (“all” is accurate if
there is no confusion) pure commitment pairs followed by mixed strategy equilibria at theaction stages lead player 1’s deviations to a single commitment feg.
To the extent of randomized commitments, however, there is an equilibrium path inwhich both players commit
111111fa; bg + fa; b; c; dg + fa; b; c; d; eg ; fag + fa; b; cg + fa; b; c; d; eg ;333333and plays (a; a) if (fa; bg ; fag), (d; a) if (fa; b; c; dg ; fag), (e; a) if (fa; b; c; d; eg ; fag), (b; b)
if (fa; bg ; fa; b; cg), (c; c) if (fa; b; c; dg ; fa; b; cg), (e; c) if (fa; b; c; d; eg ; fa; b; cg), (b; e) if(fa; bg ; fa; b; c; d; eg), (d; d) if (fa; b; c; dg ; fa; b; c; d; eg), and (e; e) if (fa; b; c; d; eg ; fa; b; c; d; eg).
The implementable strategy pro…le is 1=9 (a; a)+1=9 (d; a)+1=9 (e; a)+1=9 (b; b)+1=9 (c; c)+
1=9 (e; c) + 1=9 (b; e) + 1=9 (d; d) + 1=9 (e; e). The equilibrium payo¤ is (5; 3).
Let us demonstrate using a concrete example. If player 1 commits to fa; b; c; d; eg, herpayo¤ would be 10 in the case where player 2 commits to fag, which leads to (e; a); 4 inthe case where player 2 commits to fa; b; cg, which leads to (e; c); and 1 in the case whereplayer 2 commits to fa; b; c; d; eg, which leads to (e; e). The three cases each occur with aprobability of 1/3. Thus, player 1’s payo¤ from committing to fa; b; c; d; eg is1
1
110 +4+
1 = 5:
3
3
3
If player 1 commits to fa; b; c; dg, her payo¤ would be 10 in the case where player 2 commits
to fag, which leads to (d; a); 3 in the case where player 2 commits to fa; b; cg, which leads3to (c; c); and 2 in the case where player 2 commits to fa; b; c; d; eg, which leads to (d; d).
As above, player 1’s payo¤ from committing to fa; b; c; dg is
1
1
110 +3+
2 = 5:
3
33
If player 1 commits to fa; bg, her payo¤ would be 9 in the case where player 2 commits tofag, which leads to (a; a); 6 in the case where player 2 commits to fa; b; cg, which leadsto (b; b); and 0 in the case where player 2 commits to fa; b; c; d; eg, which leads to (b; e).
Again, player 1’s payo¤ from committing to fa; bg is
1
1
1
9+
6+
0 = 5:
3
33
If player 1’s deviation is fa; b; cg, for example, her expected payo¤ is (since each of (a; a),(c; c), and (b; e) would occur with a probability of 1/3)
1
1
19+3+0 = 4;
333
which is not pro…table. We can calculate player 2’s payo¤ in a similar way.
This outcome is interpreted as follows. If the status quo game is played, the players
play (e; e) and the payo¤ for player 1 is 1. Although (d; d) is more desirable for player 1
than (e; e), (d; d) is not played in the status quo game since player 1 has an incentive toplay e against player 2’s d. Player 1 needs to remove this temptation by eliminating e in
the …rst stage. If player 1 eliminates e, then (e; d) would be zero expectation, so that player
2 has an incentive to commit to fa; b; cg, (where in a subgame fa; b; c; dg feg, not (b; e)but (d; e) would be played,) resulting in (c; c), through which both players’payo¤s improvefrom 2 to 3 or 4. Now, the best reply for player 1 against player 2’s fa; b; cg is committing
not to fa; b; c; dg but to fa; bg, since player 2’s temptation to play d or e has disappeared.
If player 1 commits to fa:bg, then player 2’s best reply is further eliminating b and c intofag. However, against player 2’s fag , player 1 has an incentive to do nothing in the …rststage. If player 1 plays fa; b; c; d; eg, one of player 2’s best replies is doing nothing in the…rst stage. In summary, the …rst stage seems to be a matching-pennies-like situation.

3

Multiple Commitment Stages

Let us assume that the number of commitment stages increases from one to four. In then-stage commitment game of GP , denoted by n (GP ), (b; b) and (c; c) continues to be
the implementable strategy pro…le. In addition, 4 (GP ) has an implementable strategy
pro…le (a; a) that is Pareto e¢ cient. In the …rst commitment stage, player 1 eliminates ewhile player 2 does nothing. In the second commitment stage, after observing player 1’s
elimination of e, player 2 commits to fa; b; cg, followed by (c; c). Player 2’s commitment
demonstrates her temptation to play e if player 1 plays b; thus player 1 eliminates c and dwith relief, playing (b; b). By the same reasoning, player 2 can make a single commitment
to fag resulting in (a; a) in the next period.
4

When there is only one commitment stage, the best reply for player 1 to player 2’s fag
is e. However, player 1 has already eliminated e. Hence,(a; a) is realized in the action
stage. Bade et al. (2009) asserts that the number of commitment stages does not matter.
However, the class of the strategic form game with …nite strategy sets results in di¤erent
priorities. The example above and the examples in Dutta and Ishii (2016) indicate thatthe number of the commitment stages alters the outcomes of commitment games.

4No Pure Commitment Equilibria

Renou (2009) states that it considers pure strategies mainly. When players randomize theircommitments, the calculation of the lower bound of payo¤s mentioned in the main theoremchanges. To avoid the problem, a natural setting is “pure commitments and mixed actions”
as in Dutta and Ishii (2016). Unfortunately, there is a status quo game that has no suchequilibrium.
abca 5; 7 4; 8 0; 9b 1; 6 8; 5 3; 3c 13; 0 12; 2 2; 4GN
There is no subgame perfect equilibrium with pure commitments in Renou’s (2009)setting. For example, (b; b) following fa; bg fb; cg is not implementable, since player
2 has an incentive to commit to fag. Additionally, (c; c) following fa; cg fa; cg is not
implementable, since player 1 has an incentive to commit to fb; cg, followed by a mixed
strategy equilibrium that improves player 1’s payo¤. The latter case suggests how di¢ cultit is to construct a pure commitment equilibrium in (GN ).
The status quo game has the unique equilibrium (:5b + :5c; :2b + :8c). Since c dominatesa, player 1 can play a with positive probability only by eliminating c on the equilibrium
path. In truth, player 1 has an incentive to eliminate c in the status quo game, resultingin (:5a + :5b; :5a + :5b). Then, consider the commitment to fa; bg by player 1. The uniquebest reply for player 2 is a commitment to fag. Player 1’s best reply to fag is not fa; bg but
keeping option c. The above best-reply-sequence shows that (GN ) also has a matchingpennies-like structure. In addition, there is no pure commitment equilibrium.
As in the case of 4 (GP ), 2 (GN ) has a subgame perfect equilibrium outcome (a; a)by player 2 committing to fag after observing player 1 eliminating c. Still, having multiplecommitment stages is key here.

5Concluding Comments about Multiple Commitment
StagesRenou (2009) mainly considers pure strategies, and produces various results. However,there is a situation in which the mixing expansion brings Pareto improvement. Of course,
5considering pure strategies is su¢ cient in many circumstances. By contrast, a game with nopure strategy equilibrium needs mixed equilibria. In particular, an equilibrium with mixed
commitments does not tend to be intuitively comprehensive. In this paper, we consider
that including multiple commitment stages leads to “reading”from mixed equilibria to pureequilibria. When our priority is intuitive interpretation, it is better that mixed equilibria
are possible in the action stage and commitment must be pure as in Dutta and Ishii (2016).
Con‡ict of Interest: The author has no con‡ict of interest.

References[1] Sophie Bade, Guillaume Haeringer, and Ludovic Renou, 2009. Bilateral commitment. Journal of Economic Theory 144, 1817-1831.
[2] Rohan Dutta and Ryosuke Ishii, 2016. Dynamic commitment games, e¢ ciency andcoordination. Journal of Economic Theory 163, 699-727.
[3] Ludovic Renou, 2009. Commitment games. Games and Economic Behavior 66, 488505.

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