See Section 3: Paper ballots to be placed in a ballot box, counted, and made public afterwards. I think this is a fair process. It allows the votes to be made public, which should be the case, but it prevents armtwisting, machine malfunctions, and and any other mischief during the voting.

R E S O L U T I O N
BE IT RESOLVED by the House of Representatives of the State of
Texas, That, in accordance with Sections 9(b) and 11, Article III,
Texas Constitution, the House of Representatives of the 80th
Legislature shall elect a speaker of the house from its own
membership as follows:
SECTION 1. NOMINATIONS. (a) The secretary of state shall
call for nominations from the floor for the election of speaker of
the house of representatives and shall recognize every member who
desires to make a nomination. The order in which members desiring
to make a nomination shall proceed shall be determined by lot.
(b) Each member recognized for this purpose shall
immediately advance to the front microphone and make the nomination
in a nominating speech not to exceed five minutes in length.
(c) After all nominations have been made, the secretary of
state shall declare nominations to be closed.
SECTION 2. SECONDS. (a) A person is not considered a
nominee unless the nomination is seconded by at least one member.
(b) One seconding speech shall be allowed for each
nomination in the order in which nominations were made, then other
seconding speeches shall be allowed in rotation in the same order.
(c) No more than five seconding speeches shall be allowed
for each nominee.
(d) A seconding speech may not exceed three minutes in
length.
SECTION 3. VOTING PROCEDURES. (a) The chief clerk shall
prepare paper ballots containing a space for a member to write in
the member’s choice for speaker. One ballot shall be prepared for
each member and must contain the member’s name and district number
and a space for the member’s signature. The house sergeant shall
distribute each ballot to the appropriate member at the member’s
desk.
(b) The chief clerk shall designate a voting area and shall
provide a voting box for the marked ballots. A member may mark and
sign the member’s ballot either at the member’s desk or at the
voting area.
(c) Each member shall be called individually by the reading
clerk in order of the member’s district number and the member shall
proceed to the voting area when called. After arriving at the
voting area, the member shall fold the marked and signed ballot and
deposit it in the voting box.
(d) After all members have voted, the chief clerk shall
count the ballots on the house floor. The secretary of state and
the state auditor shall observe the count. A ballot may not be
counted unless it bears the signature of the member for whom it was
prepared. The chief clerk shall prepare a tally sheet indicating
how each member voted, the total votes received by each nominee, and
the total number of members who are present, not voting.
(e) All votes taken shall be record votes with the vote of
each member entered in the House Journal.
SECTION 4. RESULTS; RUNOFF. (a) The secretary of state
shall announce the results of the election. The chief clerk shall
release a copy of the tally indicating how each member voted as soon
as possible after the secretary of state announces the results.
(b) If a nominee receives a majority of the votes cast, the
secretary of state shall declare that nominee to be elected to the
office of speaker of the house of representatives. For purposes of
determining a majority of the votes cast for speaker under this
resolution, only a ballot indicating the name of a single nominee is
considered a vote cast.
(c) If no nominee receives a majority of the votes cast, the
two nominees receiving the largest number of votes in the initial
vote shall be in a runoff conducted in accordance with the voting
procedures for the initial vote. The secretary of state shall
announce the results of the runoff and declare the nominee
receiving a majority of the votes cast to be elected.
SECTION 5. TIE VOTES. (a) If in the initial vote no nominee
receives a majority of the votes cast and it is not possible to
determine the two nominees receiving the largest number of votes
cast because of a tie vote, additional rounds of voting shall occur
in the same manner as the initial vote among the same nominees,
other than a nominee who withdraws, until a nominee receives a
majority of the votes cast or, if no nominee receives a majority of
the votes cast, until it is possible to determine the two nominees
who receive the largest number of votes cast.
(b) If in a runoff between two nominees no nominee receives
a majority of the votes cast because of a tie vote, additional
rounds of runoff voting shall occur in the same manner as the
initial runoff between the same two nominees until one of the
nominees in the runoff receives a majority of the votes cast.
SECTION 6. OATH. After the election, the secretary of state
shall direct the administration of the oath of office to the
speaker-elect. The speaker shall take the chair immediately after
taking the oath of office.
SECTION 7. BALLOT RETENTION. The chief clerk shall retain
the paper ballots used in electing the speaker in a secure location.